Discrete fragmentation systems in weighted ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} ...

Lyndsay Kerr; Wilson Lamb; Matthias Langer

doi:10.1007/s00028-020-00561-6

Discrete fragmentation systems in weighted ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} ...

Kerr, Lyndsay; Lamb, Wilson; Langer, Matthias 2020-02-07 00:00:00 J. Evol. Equ. Journal of Evolution © 2020 The Author(s) Equations https://doi.org/10.1007/s00028-020-00561-6 1 1 1 Discrete fragmentation systems in weighted spaces Lyndsay Kerr , Wilson Lamb and Matthias Langer Abstract. We investigate an inﬁnite, linear system of ordinary differential equations that models the evolu- tion of fragmenting clusters. We assume that each cluster is composed of identical units (monomers), and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Addi- tionally, we establish that it is always possible to identify a weighted space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions and show that, under appropriate restrictions on the fragmentation coefﬁcients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly deﬁned exponential rate. 1. Introduction There are many diverse situations arising in nature and industrial processes where clusters of particles can merge together (coagulate) to produce larger clusters and can break apart (fragment) to produce smaller clusters. Particular examples can be found in polymer science [1,24,25], in the formation of aerosols [16] and in the powder production industry [21,23]. It is often appropriate when modelling such processes to regard cluster size as a discrete variable, with a cluster of size n,an n-mer, composed of n identical units (monomers). By scaling the mass, we can assume that each monomer has unit mass and so an n-mer has mass n. The aim is to use the mathematical model to obtain information on how clusters of different sizes evolve. In this paper, we restrict our attention to the case when no coagulation occurs, and consequently the evolution of clusters can be described by a linear, inﬁnite system of ordinary differential equations. With the number density of clusters of size n (i.e. mass n) at time t denoted by u (t ), this fragmentation system is given by Mathematics Subject Classiﬁcation: 47D06, 34G10, 80A30, 34D05 Keywords: Discrete fragmentation, Positive semigroup, Analytic semigroup, Long-time behaviour, Sobolev towers. L. Kerr et al. J. Evol. Equ. u (t ) =−a u (t ) + a b u (t ), t > 0; n n j n, j j (1.1) j =n+1 u (0) = u˚ , n = 1, 2,..., n n where a is the rate at which clusters of size n are lost, b is the rate at which clusters n n, j of size n are produced when a larger cluster of size j fragments and u˚ is the initial density of clusters of size n at time t = 0. Equation (1.1) was ﬁrst introduced in [25]to deal with the case of binary fragmentation, where it is assumed that each fragmentation event results in the creation of exactly two daughter clusters. As in [7,10,18,19], we consider the more general case, where each fragmentation event can result in the creation of two or more clusters. Since (1.1) is an inﬁnite system, it is convenient to express solutions as time-dependent sequences of the form u(t ) := (u (t )) . n=1 Throughout this paper, we need various assumptions on the fragmentation coefﬁ- cients a and b . We list these assumptions here and will refer to them in the sequel n n, j when required. Assumption 1.1. (i) For all n ∈ N, a ≥ 0. (1.2) (ii) For all n, j ∈ N, b ≥ 0 and b = 0 when n ≥ j. (1.3) n, j n, j The total mass of daughter clusters resulting from the fragmentation of a j-mer j −1 is given by nb . In most papers that have dealt with discrete fragmentation n, j n=1 systems, it is assumed that j −1 nb ≤ j for all j = 2, 3,..., (1.4) n, j n=1 i.e. there is no increase in mass at fragmentation events. If there is strict inequality in (1.4), then mass is lost by some other mechanism. However, for most of our results we do not assume that (1.4) holds; this means that mass could even be gained at fragmen- tation events. We can specify the local mass loss or mass gain with real parameters λ , j = 2, 3,..., such that j −1 nb = (1 − λ ) j, j = 2, 3,.... (1.5) n, j j n=1 In terms of the densities u (t ), the total mass of all clusters in the system at time t is given by the ﬁrst moment, M (u(t )),of u(t ), where M u(t ) := nu (t ). (1.6) 1 n n=1 1 Discrete fragmentation systems in weighted spaces A formal calculation establishes that if u is a solution of (1.1), then M u(t ) =−a u (t ) − j λ a u (t ). (1.7) 1 1 1 j j j dt j =2 The expression in (1.7) gives the rate at which mass may be lost from the system or gained and also shows that, at least formally, the total mass is conserved when a = 0 and λ = 0 for all j = 2, 3,..., i.e. when j −1 a = 0 and nb = j for all j = 2, 3,.... (1.8) 1 n, j n=1 Note that monomers cannot fragment to produce smaller clusters, and hence the case when a > 0 is interpreted as a situation in which monomers are removed from the system. In this paper, the approach we use to investigate (1.1) relies on the theory of semi- groups of bounded linear operators and entails formulating (1.1) as an abstract Cauchy problem (ACP) in an appropriate Banach space. The existence and uniqueness of solu- tions to the ACP are established via the application of perturbation results for operator semigroups. Of particular relevance is the Kato–Voigt perturbation theorem for sub- stochastic semigroups [5,22] that was ﬁrst applied to (1.1)in[18], and subsequently in similar semigroup-based investigations into (1.1), such as [8,19]. We use a reﬁned version of this theorem proved by Thieme and Voigt in [20]. In previous studies, including [18,19], the ACP associated with the fragmentation system has been formulated in the space X := f = ( f ) : f ∈ R for all n ∈ N and n| f | < ∞ . (1.9) [1] n n n n=1 n=1 Equipped with the norm f = n| f |, f ∈ X , (1.10) [1] n [1] n=1 X is a Banach space, and [1] f = M ( f ) (1.11) [1] 1 if f ∈ X is such that f ≥ 0, n ∈ N. This means that whenever u :[0, ∞) → X [1] n [1] is a non-negative solution of the fragmentation system, the norm, u(t ) ,gives the [1] total mass at time t. Other Banach spaces, with norms related to higher-order moments, have also played a prominent role [8,11], with X being replaced by X , p > 1, [1] [ p] where ∞ p X := f = ( f ) : f ∈ R for all n ∈ N and f := n | f | < ∞ . [ p] n n [ p] n n=1 n=1 (1.12) L. Kerr et al. J. Evol. Equ. Rather than restricting our investigations to spaces of the type X , we choose to [ p] work within the framework of more general weighted spaces. As we shall demon- strate, this additional ﬂexibility will enable us to establish desirable semigroup prop- erties and results that may not always be possible in an X setting. Therefore, we let [ p] w = (w ) be such that w > 0 for all n ∈ N, and deﬁne n n n=1 1 ∞ = f = ( f ) : f ∈ R for all n ∈ N and w | f | < ∞ . (1.13) n n n n n=1 n=1 Equipped with the norm f = w | f |, f ∈ , (1.14) w n n n=1 1 1 is a Banach space, which we refer to as the weighted space with weight w. Motivated by the terms in (1.1), we introduce the formal expressions ⎛ ⎞ ∞ ∞ ∞ ⎝ ⎠ A : ( f ) → (−a f ) and B : ( f ) → a b f . n n n n j n, j j n=1 n=1 n=1 j =n+1 n=1 (w) (w) 1 Operator realisations, A and B ,of A and B, respectively, are deﬁned in by (w) (w) 1 1 A f = A f, D(A ) = f ∈ : A f ∈ (1.15) w w and (w) (w) 1 1 B f = B f, D(B ) = f ∈ : B f ∈ . (1.16) w w Here, and in the sequel, D(T ) denotes the domain of the designated operator T . −1 Similarly, we shall represent the resolvent, (λI − T ) ,of T by R(λ, T ). An ACP version of (1.1), posed in the space , can be formulated as (w) (w) u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚. (1.17) Note that this reformulation of (1.1) imposes additional constraints on both the initial data and the sought solutions since we now require u˚ ∈ and also that the (w) (w) solution u(t ) ∈ D(A ) ∩ D(B ) for all t > 0. Moreover, as the derivative on the left-hand side of (1.17) is deﬁned in terms of · , it is customary to look for 1 1 1 a solution u ∈ C ((0, ∞), ) ∩ C ([0, ∞), ). Such a solution is referred to as a w w classical solution of (1.17) and has the property that u(t ) − u˚ → 0as t → 0 . (w) (w) It turns out that often, instead of using the operator A + B on the right-hand side of (1.17), one has to use its closure, which leads to the ACP (w) (w) u (t ) = (A + B )u(t ), t > 0; u(0) = u˚. (1.18) Yet another option for an operator on the right-hand side is the maximal operator, (w) G , which is deﬁned by max (w) (w) 1 1 G f = A f + B f, D G = f ∈ : A f + B f ∈ . (1.19) max max w w 1 Discrete fragmentation systems in weighted spaces However, the domain of this operator is too large in general to ensure uniqueness of solutions; see Example 4.3 and also [6] where a continuous fragmentation equation is studied. There are a number of beneﬁts to be gained by working in more general weighted spaces, least of which is the derivation of existence and uniqueness results for (1.1) in that reduce to those established in earlier X -based investigations by choosing [ p] p (w) (w) (w) w = n . For example, in Theorem 3.4 we prove that G = A + B is the generator of a substochastic C -semigroup. While this result has already been shown for the speciﬁc case w = n for p ≥ 1, see [8,18], Theorem 3.4 is formulated for more general weights and is proved by means of an alternative and novel argument that is based on theory presented in [20]. Our approach also leads to an additional invariance result, which can be used to establish the existence of solutions to the fragmentation system (1.17) for a certain speciﬁed class of initial conditions. A further major advantage of working in the more general setting of is that it yields results on the analyticity of the related fragmentation semigroups, which do not necessarily hold in the restricted case of w = n , p ≥ 1. In particular, in Theorem 5.5 we prove that, for any fragmentation coefﬁcients, we can always (w) (w) ﬁnd a weight w such that A + B is the generator of an analytic, substochastic C -semigroup on . In connection with this, it should be noted that there are no known general results that guarantee the analyticity of the fragmentation semigroup on the space X . Indeed, this provided the motivation for previous investigations into [1] fragmentation ACPs posed in higher moment spaces, which led to a sufﬁcient condition (w) (w) being found in [8]for A + B to generate an analytic semigroup on X for some [ p] p > 1. However, simple examples are also given in [8] of fragmentation coefﬁcients where the semigroup is not analytic in X for any p ≥ 1; see Example 5.6. [ p] The importance of establishing the analyticity of the semigroup associated with the fragmentation system is that analytic semigroups have extremely useful properties. (w) (w) 1 For example, if A + B generates an analytic semigroup on , then it follows immediately that the ACP (1.17) has a unique classical solution for any u˚ ∈ . In addition, when coagulation is introduced into the system, the analyticity of the (w) (w) semigroup generated by A + B can be used to weaken the assumptions that are required on the cluster coagulation rates to obtain the existence and uniqueness of solutions to the corresponding coagulation–fragmentation system of equations. Such coagulation–fragmentation systems will be considered in a subsequent publication. Once the well-posedness of the fragmentation ACP has been satisfactorily dealt with, the next question to be addressed is that of the long-term behaviour of solutions. Results on the asymptotic behaviour of solutions to (1.17)are givenin[7,12,15]for the speciﬁc case where the weight is w = n for p ≥ 1, n ∈ N. In particular, for mass-conserving fragmentation processes, where (1.8) holds, it is shown that the solution of (1.17) converges to a state where there are only monomers present if and only if a > 0 for all n ≥ 2. In Sect. 6, we continue to work with more general weights and, in the mass loss case, show that the solution of (1.17) decays to the zero state L. Kerr et al. J. Evol. Equ. over time if and only if a > 0 for all n ∈ N. This mass loss result can then be used to deduce that the solution, in the mass-conserving case, converges to the monomer state if and only if a > 0 for all n ≥ 2, this result now holding in the general weighted space . Regarding the rate at which solutions approach the steady state, the case where mass is conserved and w = n for p > 1 is examined in [12, Section 4], and it is shown that solutions decay to the monomer state at an exponential rate, which, however, is not quantiﬁed. In Sect. 6, we obtain results regarding the exponential rate of decay of solutions, both for the mass-conserving and for the mass loss cases, by working in a 1 (w) (w) space in which A + B generates an analytic semigroup. The approach we use enables us to quantify the exponential decay rate. In [19], the theory of Sobolev towers is used to investigate a speciﬁc example of (1.1) that has been proposed as a model of random bond annihilation. Of particular note is the fact that the resulting analysis provides a rigorous explanation of an apparent non- uniqueness of solutions that emanate from a zero initial condition. We shall establish that an approach involving Sobolev towers can also be used to obtain results on (1.1) , where w is for general fragmentation coefﬁcients. By writing (1.1)asanACP in (w) (w) 1 such that A + B generates an analytic, substochastic C -semigroup on ,we are able to construct a Sobolev tower and then use this to prove the existence of unique, non-negative solutions of (1.17) for a wider class of non-negative initial conditions than those in ; see Theorem 7.2. The paper is structured as follows. In Sect. 2, we provide some prerequisite results and deﬁnitions. Following this, we begin our examination of (1.1) in Sect. 3, obtaining, in particular, the aforementioned Theorem 3.4, which is then used to draw conclusions on the existence and uniqueness of solutions to (1.17) and (1.18), both in the space X and in more general spaces. We consider the pointwise system (1.1) in Sect. 4 [1] and show that for any u˚ ∈ , a solution of (1.1) can be expressed in terms of the (w) (w) (w) semigroup generated by G = A + B . We then use this result to show that (w) (w) G is a restriction of the maximal operator G . This is important in investigations max into the full coagulation–fragmentation system as it allows the fragmentation terms (w) to be completely described by the operator G . Results on the analyticity of the fragmentation semigroup are presented in Sect. 5 and then applied both in Sect. 6, where the asymptotic behaviour of solutions is investigated, and in Sect. 7, where the theory of Sobolev towers is applied to establish the well-posedness of (1.17)for more general initial conditions. 2. Preliminaries We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [9,13]. Let X be a real vector lattice with norm ·. The positive cone, X ,of X is the set of non-negative elements in X and, similarly, for a subspace D of X, we denote the set of non-negative elements in D 1 Discrete fragmentation systems in weighted spaces by D .If X is a vector lattice, then for each f ∈ X the vectors f := sup{± f, 0} are + ± well deﬁned and satisfy f , f ∈ X and f = f − f . A vector lattice, equipped + − + + − with a lattice norm ·,issaidtobea Banach lattice if X is complete under ·. Moreover, if the lattice norm satisﬁes f + g= f +g for all f, g ∈ X , then X is an AL-space. It can be shown that, when X is an AL-space, there exists a unique, bounded linear functional, φ, that extends · from X to X; see [9, Theorems 2.64 and 2.65]. We now turn our attention to C -semigroups which are crucial to our investigation into the pure fragmentation system. The notions and results given here can be found in [17]. First, we note that if (S(t )) is a C -semigroup on a Banach space X, then t ≥0 0 ωt there exist M ≥ 1 and ω ∈ R such that S(t )≤ Me for all t ≥ 0, and the growth bound, ω ,of (S(t )) is deﬁned by 0 t ≥0 ωt ω := inf ω ∈ R : there exists M ≥ 1 such that S(t )≤ M e for all t ≥ 0 . 0 ω ω Analytic semigroups, see [17, Deﬁnition II.4.5], are of particular importance in Sect. 5. Semigroups of this type have a number of useful properties that make them desirable to work with. For example, if G is the generator of an analytic semigroup, (S(t )) , on a Banach space X, then S(t ) f ∈ D(G ) for all t > 0, n ∈ N and f ∈ X, t ≥0 and S(·) is inﬁnitely differentiable. When dealing with many physical problems, such as the fragmentation system, meaningful solutions must be non-negative, and this requirement has to be taken into account in any semigroup-based investigation. In connection with this, we say that a C -semigroup (S(t )) on an ordered Banach space X, such as a Banach lattice, is 0 t ≥0 positive if S(t ) f ≥ 0 for all f ∈ X ;itiscalled substochastic (resp. stochastic)if, additionally, S(t ) f ≤ f (resp. S(t ) f = f ) for all f ∈ X . It follows that if G generates a substochastic semigroup (S(t )) , then the associated ACP t ≥0 u (t ) = Gu(t ), t > 0; u(0) = u˚, has a unique, non-negative classical solution, given by u(t ) = S(t )u˚, for any u˚ ∈ D(G) . A result on substochastic semigroups and their generators that we shall exploit is due to Thieme and Voigt [20, Theorem 2.7]. This result gives sufﬁcient conditions (w) (w) under which the closure of the sum of two operators, such as A + B in (1.17), generates a substochastic semigroup. The existence of an invariant subspace under the resulting semigroup is also established. As we demonstrate in Proposition 2.4,itis possible to adapt the Thieme–Voigt result to produce a modiﬁed version that is ideally suited for applying to the fragmentation system. We ﬁrst provide some prerequisite results that are used in the proof of this proposition. L. Kerr et al. J. Evol. Equ. Lemma 2.1. Let A be a closable operator in a Banach space X. If G = Ais the generator of a C -semigroup on X, then no other extension of A is the generator of a C -semigroup on X. Proof. Suppose that G = A and H ⊇ A are generators of C -semigroups with growth bounds ω and ω , respectively, and assume that H = G. Clearly, H ⊇ G since H is 1 2 closed. Let λ> max{ω ,ω }. Then, λ ∈ ρ(G)∩ρ(H ) and hence λI −G : D(G) → X 1 2 and λI − H : D(H ) → X are both bijective. This is a contradiction since λI − H is a proper extension of λI − G. The following lemma, which is a special case of [9, Remark 6.6], will also be used. For the convenience of the reader, we present a short proof. Lemma 2.2. Let G be the generator of a positive C -semigroup on a Banach lattice X. Then, for every f ∈ D(G), there exist g, h ∈ D(G) such that f = g − h. Proof. Let f ∈ D(G). Further, let ω be the growth bound of the semigroup generated by G,ﬁx λ>ω and set f := (λI − G) f . Since X is a Banach lattice, we have 0 0 f = f − f with f , f ∈ X .Now,let g := R(λ, G) f and h := R(λ, G) f . 0 + − + − + + − The fact that G generates a positive semigroup implies that R(λ, G) is a positive operator, and therefore g, h ∈ D(G) . Moreover, f = R(λ, G) f = R(λ, G)( f − f ) = R(λ, G) f − R(λ, G) f = g − h, 0 + − + − which proves the result. When the fragmentation coefﬁcients satisfy Assumption 1.1 and (1.8), then, as mentioned in the previous section, a formal calculation shows that the total mass is conserved. Consequently, if u is a non-negative solution of the fragmentation system, and it is known that u(t ) ∈ X for t ≥ 0, then we would expect u to satisfy [1] ∞ ∞ u(t ) = nu (t ) = nu˚ =u˚ for all t ≥ 0. [1] n [1] n=1 n=1 Clearly, this mass conservation property will hold whenever the solution can be written in terms of a stochastic semigroup on X . To this end, the following proposition will [1] prove useful. Proposition 2.3. Let (S(t )) be a positive C -semigroup on an AL-space, X, with t ≥0 0 generator G, and let φ be the unique bounded linear extension of the norm · from X to X. (i) The semigroup (S(t )) is stochastic if and only if t ≥0 φ S(t ) f = φ( f ) for all f ∈ X. (2.1) (ii) If φ(Gf ) = 0 for all f ∈ D(G) , then (2.1) holds and hence the semigroup (S(t )) is stochastic. t ≥0 1 Discrete fragmentation systems in weighted spaces (iii) Let G be an operator such that G = G .If φ(G f ) = 0 for all f ∈ D(G ) 0 0 0 0 + and each f ∈ D(G ) can be written as f = g − h, where g, h ∈ D(G ) , then 0 0 + (2.1) holds and hence (S(t )) is stochastic. t ≥0 Proof. (i) Assume that (S(t )) is stochastic and let f ∈ X and t ≥ 0. Then, t ≥0 f = f − f , where f , f ∈ X , and therefore + − + − + φ S(t ) f = φ S(t ) f − φ S(t ) f =S(t ) f −S(t ) f + − + − = f − f = φ( f ) − φ( f ) = φ( f ). + − + − Conversely, when (2.1) holds, we have S(t ) f = φ(S(t ) f ) = φ( f ) = f for f ∈ X and t ≥ 0. (ii) Let f ∈ D(G). From Lemma 2.2, there exist g, h ∈ D(G) such that f = g −h. Then, d d φ(S(t ) f ) = φ S(t ) f = φ GS(t ) f dt dt = φ GS(t )g − φ GS(t )h = 0 since S(t )g, S(t )h ∈ D(G) . Thus, φ(S(t ) f ) = φ( f ) for all f ∈ D(G), and hence also for all f ∈ X, since D(G) is dense in X. (iii) Let f ∈ D(G ). Then, f = g − h for some g, h ∈ D(G ) by assumption, and 0 0 + φ(G f ) = φ G (g − h) = φ(G g) − φ(G h) = 0. 0 0 0 0 Thus, φ(G f ) = 0 for all f ∈ D(G ).Now,let f ∈ D(G). Then, there exist 0 0 (n) (n) (n) f ∈ D(G ), n ∈ N, such that f → f and G f → Gf as n →∞. 0 0 Therefore, (n) (n) φ(Gf ) = φ lim G f = lim φ(G f ) = 0, 0 0 n→∞ n→∞ and the result follows from part (ii). We now use [20, Theorem 2.7] to obtain the following proposition, which will later be applied to the fragmentation problem. Proposition 2.4. Let (X, ·) and (Z , · ) be AL-spaces, such that (i) Z is dense in X, (ii) (Z , · ) is continuously embedded in (X, ·). Also, let φ and φ be the linear extensions of · from X to X and of · from Z + Z Z to Z, respectively. Let A : D(A) → X, B : D(B) → X be operators in X such that D(A) ⊆ D(B). Assume that the following conditions are satisﬁed. (a) −A is positive; (b) A generates a positive C -semigroup, (T (t )) ,on X; 0 t ≥0 L. Kerr et al. J. Evol. Equ. (c) the semigroup (T (t )) leaves Z invariant and its restriction to Z is a (neces- t ≥0 sarily positive) C -semigroup on (Z , · ), with generator A given by 0 Z Af = A f for all f ∈ D(A) = f ∈ D(A) ∩ Z : Af ∈ Z ; (d) B| is a positive linear operator; D(A) (e) φ((A + B) f ) ≤ 0 for all f ∈ D(A) ; (f) (A + B) f ∈ Z and φ ((A + B) f ) ≤ 0 for all f ∈ D(A) ; Z + (g) Af ≤ f for all f ∈ D(A) . Z + Then, there exists a unique substochastic C -semigroup on X which is generated by an extension, G, of A + B. The operator G is the closure of A + B. Moreover, the semigroup (S(t )) generated by G leaves Z invariant. If φ((A + B) f ) = 0 for all t ≥0 f ∈ D(A) , then (S(t )) is stochastic. + t ≥0 Proof. We ﬁrst show that the conditions of [20, Theorem 2.7] hold. From (ii) and the fact that (Z , · ) is an AL-space, it is clear that [20, Assumption 2.5] is satisﬁed. Also, from (f) and (g) we obtain that φ (A + B) f ≤ 0 ≤ f −Af Z Z for all f ∈ D(A) . Moreover, (f) and the deﬁnition of A imply that Bf ∈ Z for all f ∈ D(A) . Consequently, if we now take f ∈ D(A) and use Lemma 2.2 to express Bf as Bg − Bh, where g, h ∈ D(A) , then it follows easily that B(D(A)) ⊆ Z. Thus, all the assumptions of [20, Theorem 2.7] are satisﬁed and therefore G = A + B is the generator of a substochastic semigroup (S(t )) , which leaves Z invariant. t ≥0 That no other extension of A + B can generate a C -semigroup on X is an immediate consequence of Lemma 2.1. Finally, since A generates a substochastic C -semigroup, it follows from Lemma 2.2 that we can write any f ∈ D(A) = D(A + B) as f = g − h, where g, h ∈ D(A) . An application of Proposition 2.3 (iii) then yields the stochasticity result. 3. The fragmentation semigroup In this section, we begin our analysis of the fragmentation system (1.1) by investi- gating the associated ACP (1.17), which we recall takes the form (w) (w) u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚, (w) (w) 1 where A and B are deﬁned in by (1.15) and (1.16), respectively. A direct application of Proposition 2.4 will establish that, under appropriate conditions on the (w) (w) (w) (w) weight w, G = A + B generates a substochastic C -semigroup, (S (t )) , 0 t ≥0 1 (w) (w) 1 on . As no other extension of A + B generates a C -semigroup on ,we w w (w) 1 shall refer to (S (t )) as the fragmentation semigroup on . In the process of t ≥0 proving the existence of the fragmentation semigroup, we shall also obtain explicit 1 (w) subspaces of which are invariant under (S (t )) . t ≥0 w 1 Discrete fragmentation systems in weighted spaces First, we note that is an AL-space, with positive cone 1 ∞ 1 = f = ( f ) ∈ : f ≥ 0 for all n ∈ N , n n w n=1 w whenever w = (w ) is a positive sequence. Moreover, in this case the unique n=1 1 1 bounded linear functional, φ , that extends · from ( ) to is given by w w + w w φ ( f ) = w f for all f ∈ . (3.1) w n n n=1 We recall also that if we take w = n for all n ∈ N, then = X and · = · . n [1] w [1] (w) (w) For this speciﬁc case, we shall represent φ , A and B by M , A and B , w 1 1 1 respectively, and consequently the ACP (1.17)on X will be written as [1] u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚. (3.2) 1 1 From physical considerations, it is clear that the initial condition, u˚,inthe ACP (1.17) must necessarily be non-negative, and similarly, if u :[0, ∞) → is the cor- responding solution, then we require u(t ) to be non-negative for all t ≥ 0. Moreover, if we assume (1.4) to hold, or, equivalently, (1.5) with λ ∈[0, 1], we expect from (1.7) that mass is either lost or conserved during fragmentation. From (1.6) and the deﬁnition of the norm on X , this is equivalent to [1] u(t ) ≤u˚ for all t ≥ 0, (3.3) [1] [1] with equality being required in the mass-conserving case, provided that w is such that ⊆ X . [1] For convenience, we include the following elementary result which states that (w) 1 the operator A generates a substochastic semigroup on for any non-negative weight w. Lemma 3.1. Let and · be deﬁned by (1.13) and (1.14), respectively, and (w) let (1.2) hold. Then, the operator A , deﬁned by (1.15), is the generator of a sub- (w) 1 stochastic C -semigroup, (T (t )) ,on , which is given, for t ≥ 0, by the inﬁnite 0 t ≥0 −a t diagonal matrix diag(v (t ), v (t),...), where v (t ) = e for all n ∈ N. 1 2 n For the remainder of this section, the weight, w, will be required to satisfy the following assumption. Assumption 3.2. (i) w ≥ n for all n ∈ N. (ii) There exists κ ∈ (0, 1] such that j −1 w b ≤ κw for all j = 2, 3,.... (3.4) n n, j j n=1 L. Kerr et al. J. Evol. Equ. Remark 3.3. Let w be such that (w /n) is increasing and let (1.4) hold. Then, n=1 j −1 j −1 j −1 w w w j j w b = nb ≤ nb ≤ j = w . n n, j n, j n, j j n j j n=1 n=1 n=1 Hence, (3.4) is satisﬁed with κ = 1. In particular, if (1.4) holds, then Assumption 3.2 is automatically satisﬁed by any weight of the form w = n , p ≥ 1. (w) It is an immediate consequence of Assumption 3.2 that, for any f ∈ D(A ) ,we have ⎛ ⎞ j −1 ∞ ∞ ∞ (w) ⎝ ⎠ φ B f = w a b f = w b a f w n j n, j j n n, j j j n=1 j =n+1 j =2 n=1 (3.5) (w) ≤ κ w a f =−κφ A f . j j j w j =1 (w) Consequently, for all f ∈ D(A ), ∞ ∞ (w) (w) B f = w a b f ≤ φ B | f | w n j n, j j w (3.6) n=1 j =n+1 (w) (w) ≤−κφ A | f | = κA f , w w from which it follows that (w) (w) (w) (w) (w) (w) (w) D(A ) ⊆ D(B ) and D A + B = D(A ) ∩ D(B ) = D(A ). (3.7) (w) (w) We now apply Proposition 2.4 to the operators A and B . This involves the construction of a suitable subspace of , and to this end we require a sequence (c ) that satisﬁes n=1 c ≤ c and a ≤ c for all n ∈ N. (3.8) n n+1 n n Note that such a sequence can always be found. For example, we can take c = max{a ,..., a } for n = 1, 2,.... (3.9) n 1 n (w) Let C be the corresponding multiplication operator, deﬁned by (w) (w) 1 [C f ] =−c f , n ∈ N, D(C ) = f ∈ : w c | f | < ∞ , n n n n n n n=1 (3.10) (w) and equip D(C ) with the graph norm (w) (w) f (w) = f +C f = (w + w c )| f |, f ∈ D(C ). (3.11) w w n n n n n=1 1 Discrete fragmentation systems in weighted spaces (w) 1 ∞ Clearly, (D(C ), · (w) ) = ( , · ) with weight w = (w ) where w n w n=1 w = w + w c , n ∈ N, (3.12) n n n n and hence ( , · ) is an AL-space, and the unique linear extension of · from w w 1 1 1 ( ) to is given by φ ( f ) = w f for f ∈ . + w n n w w n=1 w ∞ ∞ We note that the choice (3.9)for (c ) is ‘maximal’ in the sense that if (cˆ ) n n n=1 n=1 is any other monotone increasing sequence that dominates (a ) and C is deﬁned n=1 (w) (w) analogously to (3.10), then D(C ) ⊆ D(C ). (w) (w) (w) Theorem 3.4. Let Assumptions 1.1 and 3.2 hold. Then, G = A + B is the (w) 1 generator of a substochastic C -semigroup, (S (t )) ,on . Moreover, the semi- 0 t ≥0 (w) (w) 1 (w) group (S (t )) leaves D(C ) = invariant, where D(C ) and w are deﬁned t ≥0 in (3.10) and (3.12), respectively, and (c ) satisﬁes (3.8). If, in addition, (1.8) holds n=1 and w = n for all n ∈ N, then the semigroup, (S (t )) , generated by G = A + B n 1 t ≥0 1 1 1 is stochastic on X . [1] Proof. We show that the conditions (i), (ii) and (a)–(g) of Proposition 2.4 are all (w) (w) satisﬁed when A = A , B = B and the AL-spaces (X, ·) and (Z , · ) are, 1 (w) 1 respectively, and (D(C ), · (w) ) = ( , · ). w C 1 1 Clearly, is dense in and continuously embedded since w ≤ w , n ∈ N.It n n follows that (i) and (ii) both hold. (w) Condition (a) is obviously satisﬁed by A , and, for (b), we apply Lemma 3.1 to (w) (w) 1 establish that A generates a substochastic C -semigroup, (T (t )) ,on .Itis 0 t ≥0 (w) 1 easy to see that the semigroup (T (t )) leaves invariant and the generator of t ≥0 1 (w) (w) 1 the restriction to is A , the part of A in ; this shows (c). w w (w) (w) It is also clear that B is positive. From (3.5), we obtain that, for f ∈ D(A ) , (w) (w) (w) (w) φ A + B f = φ (A f ) + φ (B f ) w w w (3.13) (w) (w) ≤ φ (A f ) − κφ (A f ) ≤ 0. w w Hence, (d) and (e) hold. Since w ≥ n, by Assumption 3.2 (i), we have w = w + w c ≥ n, n ∈ N. n n n n n Moreover, the monotonicity of (c ) and Assumption 3.2 (ii) imply that n=1 j −1 j −1 j −1 w b = (1 + c )w b ≤ (1 + c ) w b ≤ κ(1 + c )w = κw n n, j n n n, j j n n, j j j j n=1 n=1 n=1 for all j ∈ N. This means that Assumption 3.2 also holds for the weight w . Therefore, (w) (w) (w) (w) we obtain from (3.7) and (3.13) that D(A ) ⊆ D(B ) and φ ((A +B ) f ) ≤ 0 (w) for f ∈ D(A ) , and so (f) is also satisﬁed. That (g) holds follows from + L. Kerr et al. J. Evol. Equ. ∞ ∞ ∞ (w) A f = w a | f |≤ w c | f |≤ w | f |= f w n n n n n n n n w n=1 n=1 n=1 (w) for f ∈ D(A ) . (w) Thus, the conditions of Proposition 2.4 are all satisﬁed and therefore G = (w) (w) (w) 1 A + B is the generator of a substochastic C -semigroup, (S (t )) ,on , 0 t ≥0 (w) 1 which also leaves D(C ) = invariant. Finally, assume that (1.8) is satisﬁed and w = n for all n ∈ N. Then, equality holds in (3.4) with κ = 1 and hence also in (3.5), and so, from Proposition 2.4,the semigroup generated in this case is stochastic. Remark 3.5. Consider the case where w = n for all n ∈ N, so that = X , n [1] and let Assumption 1.1 and (1.4) hold. Then, by Remark 3.3,(3.4) is also satisﬁed, and therefore, from Theorem 3.4, the operator G = A + B is the generator of a 1 1 1 substochastic C -semigroup, (S (t )) ,on X . It follows that the ACP 0 1 t ≥0 [1] u (t ) = G u(t ), t > 0; u(0) = u˚, (3.14) with u˚ ∈ D(G ), has a unique classical solution, given by u(t ) = S (t )u˚ for all t ≥ 0. 1 1 Moreover, if u˚ ≥ 0, then this solution is non-negative. Now suppose that u˚ ∈ D(G ) 1 + and, in addition, assume that (1.8) holds. Then, the semigroup (S (t )) is stochastic 1 t ≥0 on X and so, from (1.11), [1] M u(t ) =u(t ) =S (t )u˚ =u˚ = M (u˚) for all t ≥ 0, 1 [1] 1 [1] [1] 1 showing that u(t ) is a mass-conserving solution. With the help of Remark 3.5, we obtain the following corollary. (w) (w) Corollary 3.6. Let Assumptions 1.1 and 3.2 hold and let u˚ ∈ D(G ), where G = (w) (w) A + B as in Theorem 3.4. Then, the ACP (w) u (t ) = G u(t ), t > 0; u(0) = u˚ (3.15) (w) has a unique classical solution, given by u(t ) = S (t )u˚. This solution is non-negative (w) (w) if u˚ ∈ D(G ) . Moreover, if (1.8) holds and u˚ ∈ D(G ) , then this solution is + + mass conserving. (w) Proof. It follows immediately from Theorem 3.4 that u(t ) = S (t )u˚ is the unique (w) (w) classical solution of (3.15) for all u˚ ∈ D(G ). Moreover, since (S (t )) is t ≥0 (w) substochastic, this solution is non-negative if u˚ ∈ D(G ) . (w) Now, assume that (1.8) holds and u˚ ∈ D(G ) . Then, (S (t )) is a stochastic + 1 t ≥0 C -semigroup on X . Additionally, since w ≥ n for all n ∈ N, is continuously 0 [1] n embedded in X and so, as u(t ) is differentiable in , u(t ) is also differentiable in [1] (w) 1 X and the derivatives must coincide. Moreover, since G is the part of G in , [1] 1 (w) we have u(t ) ∈ D(G ). Therefore, u(t ) = S (t )u˚ is also a solution of (3.14), and, (w) by uniqueness of solutions, it follows that S (t )u˚ = S (t )u˚ for t ≥ 0. Remark 3.5 (w) then establishes that u(t ) = S (t )u˚ is a mass-conserving solution. 1 Discrete fragmentation systems in weighted spaces (w) Note that even if u ∈ D(A ), the solution, u(t ),of (3.15) need not belong to (w) D(A ) for any t > 0. Hence, the existence of a solution of (1.17) is not guaranteed in general; one only has uniqueness of solutions. However, the next theorem shows (w) that under the stronger assumption u˚ ∈ D(C ) on the initial condition, the ACP (1.17) is well posed. (w) Theorem 3.7. Let Assumptions 1.1 and 3.2 hold. For u˚ ∈ D(C ),the ACP (1.17) (w) (w) has a unique classical solution given by u(t ) = S (t )u, ˚ t ≥ 0.If u˚ ∈ D(C ) , (w) then this solution is non-negative. Moreover, if (1.8) holds and u˚ ∈ D(C ) , then the solution is mass conserving. (w) (w) (w) (w) Proof. We know that G and A + B coincide on D(A ) and also that (w) (w) (w) u(t ) = S (t )u˚ is the unique solution of (3.15)for u˚ ∈ D(C ) ⊆ D(G ). Since (w) (w) (w) (w) (w) (S (t )) leaves D(C ) invariant, it follows that S (t )u˚ ∈ D(C ) ⊆ D(A ). t ≥0 The result then follows from Corollary 3.6. The next proposition shows that if the sequence (a ) has a certain additional n=1 (w) property, then a unique solution of (1.17)existsfor u˚ ∈ D(A ). Proposition 3.8. Let (a ) be an unbounded sequence such that (1.2) holds. Fur- n=1 ∞ ∞ ther, deﬁne the sequence (c ) by (3.9) and let w = (w ) be such that w > 0 n n n n=1 n=1 (w) (w) for all n ∈ N. Then, D(C ) = D(A ) if and only if lim inf > 0. (3.16) n→∞ Proof. Note ﬁrst that the unboundedness of (a ) implies that c →∞ as n →∞. n n n=1 (w) (w) Since c ≥ a for all n ∈ N,wehave D(C ) ⊆ D(A ).If (3.16) holds, then there n n (w) exist γ> 0, N ∈ N such that a ≥ γ c for all n ≥ N.Let f ∈ D(A ). Then, n n ∞ N −1 ∞ (w) C f = w c | f |≤ w c | f |+ w a | f | w n n n n n n n n n n=1 n=1 n=N N −1 (w) ≤ w c | f |+ A f < ∞, n n n w n=1 (w) (w) and so D(A ) = D(C ). Now, suppose that lim inf (a /c ) = 0. Then, there exists a subsequence, n→∞ n n a /c , such that n n k k k=1 a 1 1 1 c = 0, ≤ and ≤ for all k ∈ N. c k c k n n k k Let f be such that 1/(c w k) when j = n , n n k k k f = (3.17) 0 otherwise. L. Kerr et al. J. Evol. Equ. Then, ∞ ∞ ∞ 1 1 a w | f |= a w ≤ < ∞, n n n n n k k c w k k n n k k n=1 k=1 k=1 ∞ ∞ ∞ 1 1 w | f |= w ≤ < ∞, n n n c w k k n n k k n=1 k=1 k=1 ∞ ∞ c w | f |= =∞. n n n n=1 k=1 (w) (w) (w) It follows that f ∈ D(A )\D(C ), showing that D(C ) is a proper subset of (w) D(A ). Remark 3.9. If (a ) is unbounded and eventually monotone increasing, then n=1 ∞ (w) (c ) , given by (3.9), satisﬁes (3.16). Note that, in X , the invariance of D(A ) n [1] n=1 under the fragmentation semigroup has already been established in [18, Theorem 3.2] for the case when (a ) is monotone increasing. n=1 We end this section by obtaining an inﬁnite matrix representation of the fragmen- (w) 1 tation semigroup (S (t )) on , which is used in Sect. 6. Let Assumptions 1.1 t ≥0 (w) (w) (w) and 3.2 be satisﬁed so that G = A + B is the generator of a substochastic (w) 1 1 C -semigroup, (S (t )) ,on .For n ∈ N,let e ∈ be given by 0 t ≥0 n w w 1if n = k, (e ) = (3.18) n k 0 otherwise, and let (s (t )) be the inﬁnite matrix deﬁned by m,n m,n∈N (w) s (t ) = (S (t )e ) for all m, n ∈ N. m,n n m (w) Note that, since (S (t )) is positive, s (t ) ≥ 0 for all m, n ∈ N. Now, each t ≥0 m,n f ∈ can be expressed as f = f e , where the inﬁnite series is convergent n n w n=1 in . Hence, ∞ ∞ (w) (w) S (t ) f = f S (t )e = f s (t ) for all m ∈ N, n n n m,n n=1 n=1 (w) and therefore (S (t )) can be represented by the matrix (s (t )) . To deter- t ≥0 m,n m,n∈N mine s (t ) more explicitly, ﬁx n ∈ N and let (u (t), ..., u (t )) be the unique m,n 1 n solution of the n-dimensional system u (t ) =−a u (t ) + a b u (t ), t > 0; m = 1, 2,..., n; (3.19) m m j m, j j j =m+1 u (0) = 1; u (0) =0for m < n. (3.20) n m 1 Discrete fragmentation systems in weighted spaces It is straightforward to check that u(t ) = (u (t ), . . . , u (t ), 0, 0,...) solves (1.1) 1 n (w) (w) with u˚ = e . Since u(t ) ∈ D(A ) ⊆ D(G ), the function u coincides with the (w) unique solution of (3.15), and hence u(t ) = S (t )e , which yields u (t ), m = 1, 2,..., n, s (t ) = (3.21) m,n 0, m > n. For m = n, the differential equation in (3.19) reduces to u (t ) =−a u (t ), which n n −a t implies that s (t ) = u (t ) = e . Since n was arbitrary, it follows that, for all n,n n t ≥ 0, ⎡ ⎤ −a t e s (t ) s (t ) ··· 1,2 1,3 ⎢ ⎥ ⎡ ⎤ (w) −a t ⎢ ⎥ e S (t ) −a t ⎢ ⎥ (12) 0 e s (t ) ··· 2,3 (w) ⎢ ⎥ ⎣ ⎦ S (t ) = = , (3.22) ⎢ ⎥ −a t (w) ⎢ ⎥ 0 0 e ··· 0 S (t ) (22) ⎣ ⎦ . . . . . . . . . . (w) where 0 is an inﬁnite column vector consisting entirely of zeros, S (t ) is a non- (12) (w) negative inﬁnite row vector and S (t ) is an inﬁnite-dimensional, non-negative, upper (22) triangular matrix. We note that, in the particular case when = X and mass is [1] conserved, Banasiak obtains the inﬁnite matrix representation (3.22) for the semigroup (S (t )) in [7, Equation (10) and Lemma 1]. In [7], an explicit expression is also 1 t ≥0 found for s (t ), m < n, but we omit this here since it is not required for the results m,n that follow. As observed in [7, pp. 363], it follows from (3.22) that, for all N ∈ N,we have S(t ) f ∈ span{e , e ,..., e } for all f ∈ span{e , e ,..., e }. Also, note that 1 2 N 1 2 N the functions s are independent of the weight w, which implies that, whenever w is m,n (w) (w) 1 1 another weight satisfying Assumption 3.2, S (t ) and S (t ) coincide on ∩ . w w 4. The pointwise fragmentation problem and the fragmentation generator We established in Theorem 3.7 that if Assumptions 1.1 and 3.2 are satisﬁed, then (w) u(t ) = S (t )u˚ is the unique, non-negative classical solution of the fragmentation (w) ACP (1.17) for all u˚ ∈ D(C ) . Moreover, when (1.8) holds, then this solution is (w) mass conserving. Clearly, u(t ) = S (t )u˚ will also satisfy the fragmentation system (w) (1.1) in a pointwise manner when u˚ ∈ D(C ) . However, at this stage we do (w) not know in what sense, if any, the semigroup (S (t )) provides a non-negative t ≥0 solution for a general u˚ ∈ ( ) . In this section, we show that a non-negative solution of the pointwise system (1.1) can be determined for any given initial condition in 1 (w) ( ) by using the semigroup (S (t )) . + t ≥0 As before, we require Assumptions 1.1 and 3.2 to hold, and we deﬁne a sequence ∞ (w) (c ) by (3.9), with the associated multiplication operator C given by (3.10). n=1 L. Kerr et al. J. Evol. Equ. (w) (w) Then, a ≤ c for all n ∈ N and it follows that D(C ) ⊆ D(A ). From Propo- n n (w) (w) sition 3.4, D(C ) is invariant under the substochastic semigroup (S (t )) gen- t ≥0 (w) (w) (w) (w) erated by G = A + B . Consequently, u(t ) = S (t )u˚ is the unique, non- (w) negative classical solution of (1.17) for each u˚ ∈ D(C ) , and therefore t t u (t ) − u˚ =−a u (s) ds + a b u (s) ds, (4.1) n n n n j n, j j 0 0 j =n+1 for n = 1, 2,.... We use this integrated version of the pointwise fragmentation system (1.1) to prove the following result. Theorem 4.1. Let Assumptions 1.1 and 3.2 hold, and let u˚ ∈ . Then, u(t ) = (w) S (t )u˚ satisﬁes the system (1.1) for almost all t ≥ 0. Moreover, if u˚ ≥ 0, then u(t ) ≥ 0 for t ≥ 0. 1 1 1 Proof. Let u˚ ∈ ( ) and, for N ∈ N, deﬁne the operator P : → by + N w w w P f := f e = ( f , f ,..., f , 0,...), f ∈ . N n n 1 2 N n=1 (w) (N ) (w) Then, P u˚ ∈ D(C ) for all N ∈ N, and so, on setting u (t ) = S (t )P u˚,we N + N have t t (N ) (N ) (N ) u (t ) = P u˚ − a u (s) ds + a b u (s) ds, (4.2) N n n j n, j n n 0 0 j =n+1 for n = 1, 2,..., N . Clearly, P u˚ → u˚ in as N →∞, and so, by the continuity (N ) (w) of S (t ),itfollows that u (t ) → u (t ) as N →∞ for all n ∈ N and t ≥ 0. n n (N ) (N ) (w) 2 1 Moreover, if N ≥ N then u (t ) − u (t ) ≥ 0 for all t ≥ 0, since (S (t )) 2 1 t ≥0 (N ) is linear and positive. Similarly, u(t ) − u (t ) ≥ 0 for all N ∈ N and t ≥ 0. Hence, (N ) ∞ (u (t )) is monotone increasing and bounded above by u(t ), and therefore, for N =1 (N ) each ﬁxed n ∈ N, (u (t )) is monotone increasing and bounded above by u (t ). n n N =1 On allowing N →∞ in (4.2), and using the monotone convergence theorem, we obtain t t (N ) u (t ) = u˚ − a u (s) ds + lim a b u (s) ds. n n n n j n, j N →∞ 0 0 j =n+1 From this, we deduce that (N ) lim a b u (s) ds j n, j N →∞ j =n+1 exists, and a further application of the monotone convergence theorem shows that ∞ ∞ (N ) lim a b u (s) ds = a b u (s) ds. j n, j j n, j j N →∞ j =n+1 j =n+1 0 1 Discrete fragmentation systems in weighted spaces Thus, for all u ∈ ( ) , (w) (w) (w) ˚ ˚ ˚ ˚ (S (t )u) = u + −a (S (s)u) + a b (S (s)u) ds. (4.3) n n n n j n, j j j =n+1 (w) It follows that (S (t )u˚) is absolutely continuous with respect to t for each n = 1, 2,... and so (w) (w) (w) S (t )u˚ =−a S (t )u˚ + a b (S (t )u˚) , n ∈ N, (4.4) n j n, j j n n dt j =n+1 for all u˚ ∈ ( ) and almost every t ≥ 0. When u˚ is a general, and therefore not necessarily non-negative, sequence in , we can express u˚ = u˚ − u˚ ∈ . It then follows immediately from the ﬁrst part of + − (w) the proof that u(t ) = S (t )u˚ also satisﬁes (1.1) for almost all t ≥ 0. The last statement of the theorem follows immediately from the positivity of the (w) semigroup (S (t )) . t ≥0 Note that, in general, solutions of (1.1) are not unique; see the discussion in Exam- ple 4.3. We now turn our attention to obtaining a simple representation of the generator (w) (w) (w) (w) (w) G . Although we know that G coincides with A + B on D(A ), and (w) (w) also that u(t ) = S (t )u˚ is the unique classical solution of (3.15)for u˚ ∈ D(G ), (w) we have yet to ascertain an explicit expression that describes the action of G on (w) (w) D(G ). This matter is resolved by the following theorem, which shows that G (w) is a restriction of the maximal operator, G , deﬁned in (1.19). In the speciﬁc case max of X , the result has been obtained from [9, Theorem 6.20], which uses extension [ p] techniques ﬁrst introduced by Arlotti in [4] and which is applied in [8, Theorem 2.1]. We present an alternative proof, which avoids the use of such extensions. (w) Theorem 4.2. Let Assumptions 1.1 and 3.2 hold. Then, for all g ∈ D(G ), we have (w) G g =−a g + a b g , n ∈ N. (4.5) n n j n, j j j =n+1 (w) Proof. It follows from Lemma 2.2 and its proof that, for every g ∈ D(G ), there (w) (w) 1 exist g , g ∈ D(G ) such that g = g − g and f := (I − G )g ∈ ( ) for 1 2 + 1 2 j j + (w) (w) j = 1, 2. This and the linearity of G allow us to assume that g ∈ D(G ) such (w) 1 (w) that f := (I − G )g ∈ ( ) . Deﬁning u(t ) = S (t ) f ,wehavefrom (4.3) that ∞ ∞ (w) −t (w) −t R(1, G ) f = e [S (t ) f ] dt = e u (t ) dt n n 0 0 ∞ t ∞ t −t −t = f − e a u (s) ds dt + e a b u (s) ds dt. n n n j n, j j 0 0 0 0 j =n+1 L. Kerr et al. J. Evol. Equ. By Tonelli’s theorem, we have ∞ t ∞ ∞ −t −t e a u (s) ds dt = e a u (s) dt ds n n n n 0 0 0 s −s (w) = a e u (s) ds = a R(1, G ) f . n n n Using Tonelli’s theorem and the monotone convergence theorem, we obtain ∞ ∞ ∞ t −t (w) e a b u (s) ds dt = a b R(1, G ) f . j n, j j j n, j 0 0 j =n+1 j =n+1 Thus, (w) (w) (w) g = R 1, G f = f − a R 1, G f + a b R 1, G f n n n j n, j n n j j =n+1 (w) (w) (w) = I − G g − a R 1, G f + a b R 1, G f n j n, j n n j j =n+1 (w) = g − G g − a g + a b g , n n n j n, j j j =n+1 and (4.5) follows. We note that the formula (4.5) is independent of the weight w = (w ) . Being n=1 (w) able to express the action of G in this way is important when investigating the full coagulation–fragmentation system, as it enables the fragmentation terms to be (w) described by means of an explicit formula for the operator G . We shall return to this in a subsequent paper. Example 4.3. Let us consider the system u (t ) =−(n − 1)u (t ) + 2 u (t ), t > 0; n j (4.6) j =n+1 u (0) = u˚ , n = 1, 2,..., n n which coincides with (1.1)ifone sets a = n − 1, b = , n, j ∈ N, j > n. (4.7) n n, j j − 1 The system (4.6) models random scission; see, e.g. [25, equation (49)] and [14, equa- tion (10)]. It is easily seen that (1.8) is satisﬁed, and hence, mass is conserved. The example (4.6) is closely related to the example that is studied in [19, §3] and which 1 Discrete fragmentation systems in weighted spaces models random bond annihilation. More precisely, if we denote the operators for the (w) (w) (w) example from [19]by A , B , G etc., then (w) (w) (w) (w) (w) (w) A = A + I, B = B , G = G + I, (w) t (w) and hence S (t ) = e S (t ), t ≥ 0. For the particular case when w = n, n ∈ N, we have similar relations for the operators A , A etc. It follows from [19, Lemma 3.6] 1 1 that every λ> 0 is an eigenvalue of the maximal operator G [i.e. the operator 1,max (w) (λ) (λ) G deﬁned in (1.19)for w = n] with eigenvector g = (g ) where n n n∈N max (λ) g = , n ∈ N. (4.8) (λ + n − 1)(λ + n)(λ + n + 1) The existence of positive eigenvalues of G implies that G is a proper exten- 1,max 1,max sion of G . Note that the domain of G is determined explicitly in [19, Theorem 3.7], 1 1 from which we obtain that D(G ) = f = ( f ) ∈ D(G ) : lim n f = 0 . (4.9) 1 k k∈N 1,max k n→∞ k=n+1 (λ) Using the eigenvectors g from (4.8), we can deﬁne the function (λ) λt (λ) u (t ) := e g , t ≥ 0, which is a solution of the ACP u (t ) = G u(t ), t > 0; u(0) = u˚ (4.10) 1,max (λ) with u˚ = g . On the other hand, since the semigroup (S (t )) is analytic by [19, 1 t ≥0 (λ) Theorem 3.4], the function u(t ) = S (t )g , t ≥ 0, is also a solution of (4.10) and (λ) is distinct from u . This shows that, in general, one does not have uniqueness of solutions of the ACP, (4.10), corresponding to the maximal operator, G , and 1,max hence, also solutions of (1.1) are not unique. (w) More generally, a speciﬁc characterisation of D(G ) is given by Banasiak and Arlotti [9, Theorem 6.20], but this does not lead to an explicit description, such as that obtained in Example 4.3. 5. Analyticity of the fragmentation semigroup In Sect. 3, we established that Assumptions 1.1 and 3.2 are sufﬁcient condi- (w) (w) (w) tions for G = A + B to be the generator of a substochastic C -semigroup, (w) 1 (S (t )) ,on . This enabled us to obtain results on the existence and uniqueness t ≥0 (w) of solutions to (1.17). We now investigate the analyticity of (S (t )) and prove t ≥0 that, given any fragmentation coefﬁcients, it is always possible to construct a weight, (w) (w) w, such that A + B is the generator of an analytic, substochastic C -semigroup 0 L. Kerr et al. J. Evol. Equ. on . This particular result, which is one of the main motivations for carrying out an analysis of the fragmentation system in general weighted spaces, requires a stronger assumption on the weight w. Note that when dealing with analytic semigroups, we use complex versions of the spaces . Assumption 5.1. (i) w ≥ n for all n ∈ N. (ii) There exists κ ∈ (0, 1) such that j −1 w b ≤ κw for all j = 2, 3,.... (5.1) n n, j j n=1 Note that Assumption 5.1 is obtained from Assumption 3.2 by simply replacing κ ∈ (0, 1] with κ ∈ (0, 1). By removing the possibility of κ = 1, we can obtain the following improved version of Theorem 3.4. (w) (w) Theorem 5.2. Let Assumptions 1.1 and 5.1 hold. Then, the operator G = A + (w) (w) 1 B is the generator of an analytic, substochastic C -semigroup, (S (t )) ,on . 0 t ≥0 (w) (w) Proof. Let (T (t )) be as in Lemma 3.1.For α> 0 and f ∈ D(A ) , we obtain t ≥0 + from (3.6) that ! ! (w) (w) ! ! B T (t ) f dt ! ! (w) (w) ! ! ≤ κ A T (t ) f dt α α (w) (w) (w) (w) = κ φ −A T (t ) f dt = κφ − A T (t ) f dt w w 0 0 (w) (w) = κφ − T (t ) f dt = κφ f − T (α) f w w dt (w) = κ f − κT (α) f ≤ κ f . w w w (w) (w) (w) Since κ< 1, it follows from [20, Theorem A.2] that G = A + B is the generator of a positive C -semigroup. The proof of [20, Theorem A.2] establishes (w) that this semigroup is substochastic since κ< 1. Moreover, by Lemma 3.1, A is (w) 1 also the generator of a substochastic C -semigroup, (T (t )) ,on , and a routine 0 t ≥0 calculation shows that ! ! 1 1 (w) ! ! R(λ, A ) f = w | f |≤ f ,λ ∈ C\R with Re λ> 0, n n w |λ + a | | Im λ| n=1 1 (w) for all f ∈ . Therefore, by [17, Theorem II.4.6], (T (t )) is an analytic semi- t ≥0 (w) (w) (w) group. Also, the positivity of (S (t )) implies that A +B is resolvent positive. t ≥0 (w) Hence, by [2, Theorem 1.1], (S (t )) is analytic. t ≥0 1 Discrete fragmentation systems in weighted spaces Remark 5.3. (i) Although Assumption 5.1 is never satisﬁed when (1.8) holds and w = n for all n ∈ N, this does not rule out the possibility of an analytic fragmentation semigroup on X existing. Indeed, the semigroup (S (t )) in [1] 1 t ≥0 Example 4.3 is analytic, which follows from [19, Theorem 3.4] as mentioned above. (ii) If there exists λ > 0 such that (1.5) holds with λ ≥ λ for all j ≥ 2(which 0 j 0 corresponds to a ‘uniform’ mass loss case), then Assumption 5.1 immediately holds with w = n for all n ∈ N, and κ = 1 − λ . n 0 The following lemma gives sufﬁcient conditions under which Assumption 5.1 holds. Lemma 5.4. Let w be such that w w n n+1 1 ≤ ≤ δ for all n ∈ N, (5.2) n n + 1 where δ ∈ (0, 1). Moreover, let (1.4) hold. Then, Assumption 5.1 is satisﬁed with κ = δ. Proof. Since w w w n j j j −n ≤ δ ≤ δ for all n = 1,..., j − 1, n j j it follows that j −1 j −1 j −1 w w n j w b = nb ≤ δ nb ≤ δw n n, j n, j n, j j n j n=1 n=1 n=1 for j = 2, 3,..., where (1.4) is used to obtain the last inequality. Since δ ∈ (0, 1),the result follows immediately. This leads to the main result of this section. Theorem 5.5. For any given fragmentation coefﬁcients for which Assumption 1.1 ∞ (w) (w) holds, we can always ﬁnd a weight, w = (w ) , such that A + B is the n=1 generator of an analytic, substochastic C -semigroup on . If, in addition, (1.4) holds, we can choose w = r with arbitrary r > 2 and κ = 2/r so that (5.1) holds. Proof. For the ﬁrst statement, note that we can choose w ≥ n iteratively so that (5.1) is satisﬁed. The claim then follows from Theorem 5.2. Now, assume that (1.4) holds. Let r > 2, w = r for n ∈ N, and δ = 2/r, which satisﬁes δ< 1. Then, w ≥ n and n+1 n n n w 2 r 2r 2r r w n+1 n δ = · = ≥ = = , n + 1 r n + 1 n + 1 n + n n n which shows that (5.2) is satisﬁed. Hence, Lemma 5.4 implies that Assumption 5.1 is fulﬁlled. L. Kerr et al. J. Evol. Equ. As mentioned earlier, analytic semigroups have a number of desirable properties, and Theorem 5.5 will play an important role when we investigate the full coagulation– fragmentation system in a subsequent paper. In particular, Theorem 5.5 will enable us to relax the usual assumptions that are imposed on the coagulation rates in order to obtain the existence and uniqueness of solutions to the full coagulation–fragmentation system. It should be noted that a condition that is equivalent to Assumption 5.1 has previously been used as a condition for analyticity in the mass-conserving case by Banasiak; see [8, Theorem 2.1]. However, the choice of weights in [8] is restricted to w = n , p > 1, and Assumption 5.1 need not be satisﬁed for these weights for any p > 1as the following example shows. Example 5.6. Consider the mass-conserving case where a cluster of mass n breaks into two clusters, with respective masses 1 and n −1. The corresponding fragmentation coefﬁcients take the form b = 2; b = b = 1, j ≥ 3; b = 0, 2 ≤ n ≤ j − 2. (5.3) 1,2 1, j j −1, j n, j For the choice a = 0; a = n, n ≥ 2; w = n , n ∈ N; p ≥ 1, 0 n n it is proved in [7, Theorem 3] (for p = 1) and [8, Theorem A.3] (for p > 1) that (w) the semigroup generated by G is not analytic. On the other hand, Theorem 5.5 (w) guarantees the existence of exponentially growing weights w such that G = (w) (w) A + B generates an analytic semigroup. It is easy to show that for this particular example one can also choose powers of 2, namely w = 1 and w = 2 for n ≥ 2, in 1 n which case κ = 5/8. 6. Asymptotic behaviour of solutions There have been several earlier investigations into the long-term behaviour of solu- tions to the mass-conserving fragmentation system (1.1), when (1.8) holds. In partic- ular, the case of mass-conserving binary fragmentation is dealt with in [15], where it is shown that, under suitable assumptions, the unique solution emanating from u˚ must converge in the space X to the expected steady-state solution M (u˚)e , where [1] 1 1 M (u˚) and e are given by (1.6) and (3.18), respectively. This was followed by [7], and 1 1 [12] where, once again, the expected long-term steady-state behaviour is established, but now for the mass-conserving multiple fragmentation system. More speciﬁcally, in [7], a semigroup-based approach is used to prove that, for any u˚ ∈ X , [1] lim S (t )u˚ − M (u˚)e = 0 if and only if a > 0 for all n = 2, 3,.... 1 1 [1] n t →∞ 1 Discrete fragmentation systems in weighted spaces That the corresponding result is also valid in the higher moment spaces X , p > 1, [ p] is established in [12], and, under additional assumptions on the fragmentation coefﬁ- cients, it is shown in [12, Theorem 4.3] that there exist constants L > 0 and α> 0 such that the fragmentation semigroup (S (t )) on X , p > 1, satisﬁes p t ≥0 [ p] −αt S (t )u˚ − M (u˚)e ≤ Le u˚ , (6.1) p 1 1 [ p] [ p] for all u˚ ∈ X . It follows from [3] that the fragmentation semigroup (S (t )) has the [ p] p t ≥0 asynchronous exponential growth (AEG) property (with λ = 0in [3, equation (3)], i.e. with trivial growth). The assumptions required in [12] to prove that (6.1) holds in some X space are somewhat technical and not straightforward to check. Moreover, [ p] no information on the size of the constant α, and hence the exponential rate of decay to the steady state is provided. Our aim in this section is to address these issues. Working within the framework of more general weighted spaces, we study the long-term dynamics of solutions in both the mass-conserving and mass loss cases. When mass is conserved, we establish simpler conditions under which the fragmentation semigroup (w) 1 (S (t )) satisﬁes an inequality of the form (6.1) on some space , and also t ≥0 quantify α. We begin by considering the general fragmentation system (1.1), where the coefﬁ- (w) (w) (w) cients a and b satisfy Assumption 1.1, and recall that G = A + B is the n n, j (w) 1 generator of a substochastic C -semigroup, (S (t )) ,on whenever Assump- 0 t ≥0 (w) (w) (w) tion 3.2 holds. Furthermore, (S (t )) is analytic, with generator A + B when t ≥0 the more restrictive Assumption 5.1 is satisﬁed. Theorem 6.1. Let Assumptions 1.1 and 3.2 hold. (i) Then, (w) lim S (t )u˚ =0(6.2) t →∞ for all u˚ ∈ if and only if a > 0 for all n ∈ N. (ii) If, additionally, we choose w such that Assumption 5.1 is satisﬁed, and set a := inf a , then n∈N (w) −(1−κ)a t S (t )≤ e , (6.3) and hence, if a > 0 and α ∈[0,(1 − κ)a ), we have 0 0 αt (w) 1 lim e S (t )u˚ = 0 for every u˚ ∈ . (6.4) t →∞ If α> a , then (6.4) does not hold. In particular, if a = 0, then (6.4) does not 0 0 hold for any α> 0. Proof. (i) First assume that a > 0 for all n ∈ N.Let u ∈ , and, as in Sect. 4,let P u˚ = (u˚ , u˚ ,..., u˚ , 0,...), N ∈ N. For each ﬁxed n ∈ N, we know from (3.22) N 1 2 N (w) that (S (t )e ) = s (t ) = 0for m > n. Furthermore, (s , s ,..., s ), with n m m,n 1,n 2,n n,n the identiﬁcation (3.21), is the unique solution of the n-dimensional system (3.19). Our assumption on the coefﬁcients a means that all eigenvalues of the matrix associated n L. Kerr et al. J. Evol. Equ. with (3.19) are negative. It follows that s (t ) → 0as t →∞ for m = 1,..., n, and m,n therefore (w) lim S (t )e = lim w s (t ) = 0, n w m m,n t →∞ t →∞ m=1 for all n ∈ N. This in turn implies that (w) (w) S (t )P u˚ ≤ |u˚ |S (t )e →0as t →∞, N w n n w n=1 for each N ∈ N. Given any ε> 0, we can always ﬁnd N ∈ N and t > 0 such that ε ε (w) u˚ − P u˚ < and S (t )P u˚ < for all t ≥ t . N w N w 0 2 2 Then, ! ! (w) (w) (w) ! ! S (t )u˚ ≤ S (t ) u˚ − P u˚ +S (t )P u˚ w N N w (w) ≤u˚ − P u˚ +S (t )P u˚ <ε for all t ≥ t , N w N w 0 which establishes (6.2). On the other hand, suppose that a = 0for some N ∈ N. Then, we have that the (w) ∞ unique solution of (1.17), with u˚ = e ,is u(t ) = S (t )e = (s (t )) . Since N N m,N m=1 −a t s (t ) = e = 1, it is clear that u(t ) → 0as t →∞. N ,N 1 (w) (w) (ii) Now, let Assumption 5.1 hold and let u˚ ∈ ( ) . From Theorem 5.2, A +B (w) 1 generates an analytic, substochastic C -semigroup, (S (t )) ,on , and u(t ) = 0 t ≥0 (w) S (t )u˚ is the unique, non-negative classical solution of (1.17). Let t > 0. Using (3.5), we obtain that (w) (w) φ u(t ) = φ u (t ) = φ A u(t ) + φ B u(t ) w w w w dt (w) (w) ≤ φ A u(t ) − κφ A u(t ) w w =−(1 − κ) w a u (t ) n n n n=1 ≤−(1 − κ)a φ u(t ) . 0 w Therefore, −(1−κ)a t (w) −(1−κ)a t 0 0 φ u(t ) ≤ φ (u˚)e and hence S (t )u˚ ≤ e u˚ , w w w w (w) and (6.3) then follows from the positivity of (S (t )) and [9, Proposition 2.67]. If t ≥0 a > 0 and α ∈[0,(1 − κ)a ), then (6.4) holds. 0 0 On the other hand, if we choose α> a , then there exists N ∈ N such that a <α, 0 N (w) −a t −αt in which case (S (t )e ) = e > e for t > 0, and so N N αt (w) αt (w) αt −αt e S (t )e ≥ e w S (t )e > e w e = w . N w N N N N Hence, (6.4) cannot hold for any α> a . 0 1 Discrete fragmentation systems in weighted spaces Remark 6.2. When the assumptions of Theorem 6.1 are satisﬁed and a > 0 for all n ∈ N, then (6.2) shows that the only equilibrium solution of (1.17)is u(t ) ≡ 0, and this equilibrium is a global attractor for the system. On the other hand, if a = 0for at least one n ∈ N, then u(t ) ≡ 0 is not a global attractor. We now examine the mass-conserving case and assume that (1.8) holds. Note that, in this mass-conserving case, the fragmentation semigroup (S (t )) is stochastic 1 t ≥0 on the space X . Our aim is to establish an version of the results obtained in [1] (w) [7,12,15]. To this end, we recall the matrix representation of S (t ) given by (3.22) (w) and also deﬁne a sequence space Y , and its norm · (w),by (w) ∞ ∞ 1 Y = f = ( f ) : f = ( f ) ∈ and f = w | f |, (w) n n n n n=2 n=1 w Y n=2 (w) 1 1 respectively. Clearly, Y is a weighted space and can be identiﬁed with , where (w) 1 w = w for n ∈ N. Moreover, we deﬁne the embedding operator J : Y → n n+1 by Jf = (0, f , f ,...) for all f ∈ . 2 3 1 ∞ Lemma 6.3. Let α ≥ 0 and f ∈ be ﬁxed, and deﬁne f := ( f ) .IfAssump- n=2 tions 1.1, 3.2 and (1.8) hold, then (w) (w) (w) ˜ ˜ S (t ) f ≤S (t ) f − M ( f )e ≤ (w + 1)S (t ) f (6.5) (w) (w) 1 1 w 1 Y Y (22) (22) for all t ≥ 0. Proof. It follows from (3.22) that (w) (w) (w) ˜ ˜ S (t ) f = f + S (t ) f e + JS (t ) f . (6.6) 1 1 (12) (22) From this, we deduce that (w) (w) (w) ˜ ˜ S (t ) f − M ( f )e = w f + S (t ) f − M ( f ) +S (t ) f (6.7) (w) 1 1 w 1 1 1 (12) (22) and so (w) (w) S (t ) f − M ( f )e ≥S (t ) f (w) , 1 1 w (22) which is the ﬁrst inequality in (6.5). On the other hand, from Proposition 2.3 (i) and the stochasticity of (S (t )) on 1 t ≥0 (w) X , we know that M (S (t ) f ) = M (S (t ) f ) = M ( f ).Using (6.6), we obtain [1] 1 1 1 1 L. Kerr et al. J. Evol. Equ. (w) (w) ˜ ˜ f + S (t ) f − M ( f ) = M ( f ) − M f + S (t ) f e 1 1 1 1 1 1 (12) (12) (w) (w) = M S (t ) f − M f + S (t ) f e 1 1 1 1 (12) (w) (w) ≤ M S (t ) f − f + S (t ) f e 1 1 1 (12) (w) (w) ≤ φ S (t ) f − f + S (t ) f e w 1 1 (12) (w) = φ (JS (t ) f (22) ! ! (w) ! ˜! = S (t ) f . (w) (22) Y The second inequality in (6.5) then follows from (6.7). We are now in a position to prove the main theorem of this section. The ﬁrst part (w) 1 1 conﬁrms that S (t )u˚ → M (u˚)e in as t →∞, for all u˚ ∈ , provided that 1 1 w w Assumption 3.2 holds and the fragmentation rates, a , are positive for all n ≥ 2. In the second part, which deals with quantifying the rate of convergence to equilibrium, the fragmentation coefﬁcients are assumed additionally to be bounded below by a positive constant, and Assumption 3.2 is strengthened to Assumption 5.1. In this case, (w) the decay to zero of S (t )u˚ − M (u˚)e is shown to occur at an exponential 1 1 w rate, deﬁned explicitly in terms of the rate coefﬁcients and the constant κ ∈ (0, 1) in Assumption 5.1. Theorem 6.4. Let Assumptions 1.1 and 3.2, and (1.8) hold and let M be as in (1.6). (i) We have (w) lim S (t )u˚ − M (u˚)e = 0 (6.8) 1 1 w t →∞ for all u˚ ∈ if and only if a > 0 for all n ≥ 2. (ii) Choose w such that Assumption 5.1 holds and let a := inf a . Then, for 0 n∈N:n≥2 n all u˚ ∈ , (w) −(1−κ) a t S (t )u˚ − M (u˚)e ≤ (w + 1)e u˚ , (6.9) 1 1 w 1 w and so αt (w) lim e S (t )u˚ − M (u˚)e = 0, (6.10) 1 1 w t →∞ whenever a > 0 and α ∈[0,(1 − κ) a ). 0 0 Equation (6.10) does not hold for any α> a . In particular, if a = 0, then 0 0 (6.10) does not hold for any α> 0. Proof. Removing the equation for u from (1.1) leads to a reduced fragmentation (w) 1 system that can be formulated as an ACP in Y = , where, as before, w = w n n+1 for all n ∈ N. The fragmentation coefﬁcients, ( a ) and (b ) , associated n n, j n, j ∈N:n< j n=1 with the reduced system are given by a = a and b = b . Clearly, a ≥ 0 n n+1 n, j n+1, j +1 n 1 Discrete fragmentation systems in weighted spaces and b ≥ 0 for all n, j ∈ N and b = 0if n ≥ j, and w = w ≥ n + 1 > n for n, j n, j n n+1 all n ∈ N. Moreover, for j = 2, 3,..., j −1 j −1 j j w b = w b = w b ≤ w b n n, j n+1 n+1, j +1 k k, j +1 k k, j +1 n=1 n=1 k=2 k=1 ≤ κw = κw . j +1 j Hence, Assumptions 1.1 and 3.2 are satisﬁed by w , a and b , and it follows from n n, j Theorem 3.4 and (3.22) that associated with the reduced system is a substochastic (w) C -semigroup on Y , which can be represented by the inﬁnite matrix ⎡ ⎤ ⎡ ⎤ − a t −a t 1 2 e s (t ) s (t ) ··· e s (t ) s (t ) ··· 1,2 1,3 1,2 1,3 − a t −a t ⎢ 2 ⎥ ⎢ 3 ⎥ 0 e s (t ) ··· 0 e s (t ) ··· 2,3 2,3 ⎢ ⎥ ⎢ ⎥ ⎢ − a t ⎥ = ⎢ −a t ⎥ , (6.11) 3 4 00 e ··· 00 e ··· ⎣ ⎦ ⎣ ⎦ . . . . . . . . . . . . . . . . . . . . . . . . where, for all n ∈ N, m = 1,..., n − 1, t ≥ 0, s (t ) is the unique solution of m,n s (t ) =− a s (t ) + a b s (t ) m m,n j m, j j,n m,n j =m+1 n+1 =−a s (t ) + a b s (t ). m+1 m,n k m+1,k k−1,n k=m+2 An inspection of (3.19), together with (3.21), shows that s (t ) = s (t ) for all m,n m+1,n+1 (w) n ∈ N, m = 1,..., n − 1, t ≥ 0, and therefore the substochastic semigroup on Y is (w) (w) given by (S (t )) , where (S (t )) is the inﬁnite matrix that features in (3.22). t ≥0 t ≥0 (22) (22) (i) Let u˚ = (u˚ , u˚ ,...) for each u˚ ∈ . From Theorem 6.1, we deduce that 2 3 ! ! ! ! (w) (w) ! ! ! ! lim S (t )u˚ = lim S (t )u˚ = 0, (w) (22) (22) Y w t →∞ t →∞ if and only if a > 0 for all n ≥ 2, and the result is then an immediate consequence of Lemma 6.3. (ii) The calculations above show that, when Assumption 3.2 holds for w and the coefﬁcients (b ), it is also satisﬁed by w and (b ) with exactly the same value of n, j n, j κ. Therefore, from Theorem 6.1, (w) −(1−κ) a t S (t )≤ e , (22) and (6.9) follows immediately from Lemma 6.3. Moreover, if a > 0 and α ∈ [0,(1 − κ) a ), then we obtain (6.10). If α> a , then, from Theorem 6.1,the result ! ! ! ! (w) (w) αt αt ! ! ! ! lim e S (t )u˚ = lim e S (t )u˚ = 0, (w) (22) (22) Y w t →∞ t →∞ does not hold for all u˚ ∈ . Hence, from Lemma 6.3,(6.10) does not hold if α> a . 0 L. Kerr et al. J. Evol. Equ. Remark 6.5. When the assumptions of Theorem 6.4 are satisﬁed, then it follows from (3.22) that u = Me is an equilibrium solution of the mass-conserving fragmentation M 1 system for all M ∈ R. In addition, the basin of attraction for u is given by {u˚ ∈ : M (u˚) = M } provided that the assumptions of Theorem 6.4 hold and a > 0 for all 1 n n ≥ 2. On the other hand, if a = 0for some N ≥ 2, then Me is also an equilibrium N N solution for every M ∈ R. 7. Sobolev towers In this section, we use a Sobolev tower construction to obtain existence and unique- ness results relating to the pure fragmentation system for a larger class of initial con- ditions. Sobolev towers appear to have been ﬁrst applied to the discrete fragmentation system (1.1)in[19], where the authors examine a speciﬁc example and use Sobolev towers to explain an apparent non-uniqueness of solutions. As we demonstrate below, the theory of Sobolev towers is applicable to more general fragmentation systems and, in the following, the only restrictions that are imposed are that the fragmentation coef- ﬁcients satisfy Assumption 1.1, and also that a weight, w = (w ) , has been chosen n=1 (w) (w) (w) so that Assumption 5.1 holds. These restrictions imply that G = A + B is the (w) 1 generator of an analytic, substochastic C -semigroup, (S (t )) ,on .Let ω be 0 t ≥0 0 (w) (w) the growth bound of (S (t )) . Choosing μ>ω , we rescale (S (t )) to obtain t ≥0 0 t ≥0 (w) −μt (w) an analytic semigroup, (S (t )) = (e S (t )) , with a strictly negative t ≥0 t ≥0 (w) (w) (w) (w) 1 growth bound. The generator of (S (t )) is G = G −μI.Weset X = , t ≥0 (w) (w) (w) (w) (w) (w) · := · , S (t ) = S (t ), S (t ) = S (t ), and G = G . 0 w 0 0 0 (w) As described in [17, §II.5(a)], (S (t )) can be used to construct a Sobolev t ≥0 (w) tower, (X ) ,via n n∈N ! ! (w) (w) n (w) n (w) n ! ! X := D (G ) , · ; f = (G ) f , f ∈ D (G ) , n ∈ N. n n (w) For each n ∈ N, X is referred to as the Sobolev space of order n associated with the (w) (w) (w) (w) (w) semigroup (S (t )) . We also deﬁne the operator G : X ⊇ D(G ) → X t ≥0 n n n n (w) to be the restriction of G to (w) (w) (w) (w) (w) n+1 (w) D(G ) = f ∈ X : G f ∈ X = D (G ) = X , n n n n+1 for each n ∈ N. Sobolev spaces of negative order, −n, n ∈ N, are deﬁned recursively by ! ! (w) (w) (w) (w) −1 ! ! X = X , · ; f = (G ) f , f ∈ X , −n −n −n −n+1 −n+1 −n+1 −n+1 (7.1) where (X, ·) denotes the completion of the normed vector space (X, ·). Oper- (w) ators G can then be obtained in a similar recursive manner for each n ∈ N, −n (w) (w) (w) (w) with G deﬁned as the unique extension of G from D(G ) = X to −n −n+1 −n+1 −n+2 (w) (w) D(G ) = X ;see [17, §II.5(a)]. −n −n+1 1 Discrete fragmentation systems in weighted spaces (w) From [17, §II.5(a)], it follows that G is the generator of an analytic, substochastic (w) (w) (w) C -semigroup, (S (t )) ,on X for all n ∈ Z, where S (t ) is the unique, 0 n t ≥0 n −n (w) (w) (w) continuous extension of S (t ) from X to X for each t ≥ 0 and n ∈ N. Since −n (w) −μt (w) S (t ) = e S (t ), we also obtain the analytic, substochastic C -semigroup, (w) (w) (w) (w) μt (S (t )) , deﬁned on X by S (t ) = e S (t ). More generally, it is known t ≥0 −n −n −n −n (w) (w) (w) (w) that S (t ) is the unique, continuous extension of S (t ) from X to X when n m m n (w) (w) m, n ∈ Z with m ≥ n. The analyticity of (S (t )) on X , also enables us to n t ≥0 n prove the following key result. (w) (w) (w) Lemma 7.1. Let u˚ ∈ X for some ﬁxed n ∈ Z. Then, S (t )u˚ ∈ X for all n n m m ≥ n and t > 0. (w) (w) (w) Proof. It is obvious that S (t )u˚ ∈ X for all t ≥ 0 and u˚ ∈ X . Also, if n n n (w) (w) S (t )u˚ ∈ X for some m ≥ n and all t > 0, then, on choosing t ∈ (0, t ),wehave n m 0 (w) (w) (w) (w) (w) S (t )u˚ = S (t − t )S (t )u˚ ∈ D(G ) = X , 0 0 n m n m m+1 (w) (w) (w) where we have used the fact that S (t ) and S (t ) coincide on X together with n m m (w) the analyticity of S (t ). The result then follows by induction. We can now prove the following result regarding the solvability of (1.17). Theorem 7.2. Let Assumptions 1.1 and 5.1 hold. Further, let n ∈ N. Then, the ACP (w) 1 1 (1.17) has a unique, non-negative solution u ∈ C ((0, ∞), ) ∩C ([0, ∞), X ) for w −n (w) (w) all u˚ ∈ (X ) . This solution is given by u(t ) = S (t )u˚, t ≥ 0. −n −n (w) (w) μt Proof. Let u˚ ∈ (X ) and let u(t ) = S (t )u˚ = e v(t ), t ≥ 0, where v(t ) = −n −n (w) (w) (w) S (t )u˚. Then, v ∈ C ((0, ∞), X ) ∩ C ([0, ∞), X ) is the unique classical −n −n −n solution of (w) v (t ) = G v(t ), t > 0; v(0) = u˚. (7.2) −n (w) (w) (w) (w) Also, from Lemma 7.1, S (t )u˚ ∈ X = D(G ) for all t > 0. Since (S (t )) t ≥0 −n −n (w) (w) coincides with (S (t )) on D(G ),itfollows that t ≥0 (w) (w) (w) S (t )u˚ = S (t − t )S (t )u˚, where t ∈ (0, t ). 0 0 0 −n −n Consequently, (w) (w) (w) (w) (w) (w) S (t )u˚ = G S (t − t )S (t )u˚ = G S (t )u˚, t > 0, 0 0 −n −n −n dt (w) where the derivative is with respect to the norm on X = . This establishes that (w) 1 1 u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ) and also that u satisﬁes (1.17). The non- w −n negativity of u follows from the substochasticity of the semigroups. For uniqueness, we observe ﬁrst that the construction of the Sobolev tower ensures (w) (w) (w) (w) that X is continuously embedded in X . Moreover, G is the restriction of G −n −n 0 L. Kerr et al. J. Evol. Equ. (w) (w) (w) 1 1 to X = D(G ). Consequently, if u , u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ) 1 2 w −n −μt both satisfy (1.17), and we set v (t ) = e u (t ), i = 1, 2, then the difference v − v i i 1 2 is the unique classical solution of (7.2) with u˚ = 0, and so v = v , from which it 1 2 follows that u = u . 1 2 Finally, we make the following remark on the solvability of (3.2). Remark 7.3. For ﬁxed n ∈ N, the previous theorem establishes that the ACP (1.17) (w) 1 1 has a unique, non-negative solution u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ),given w −n (w) (w) ˚ ˚ by u(t ) = S (t )u, for all u ∈ (X ) , provided that Assumptions 1.1 and 5.1 −n −n are satisﬁed. Recalling that we also assume that w ≥ n for all n ∈ N,wehave that is continuously embedded in X , and from this we deduce that if u(t ) is [1] differentiable with respect to the norm on then it is also differentiable with respect to the norm on X , and the derivatives coincide. Since A + B is an extension of [1] 1 1 (w) (w) (w) (w) G = A + B , we conclude that u(t ) = S (t )u˚ also satisﬁes (3.2). −n Acknowledgements L. Kerr gratefully acknowledges the support of The Carnegie Trust for the Univer- sities of Scotland. All authors would like to thank the referees for their very helpful comments. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/ by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. REFERENCES [1] M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys. 65 (1979), 203–230 [2] W. Arendt and A. Rhandi, Perturbation of positive semigroups. Arch. Math. (Basel) 56 (1991), 107–119 [3] O. Arino, Some spectral properties for the asymptotic behavior of semigroups connected to popu- lation dynamics. SIAM Rev. 34 (1992), 445–476 [4] L. 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Nhumaio, S. T. Sonne, M. Gunsing, J. Straatsma, M. Verschueren, M. Sibeijn, G. Schulte, U. Fritsching, K. Bauckhage, C. Tropea, M. Sommerfeld, A. P. Watkins, A. J. Yule and H. Schønfeldt, Simulation of agglomeration in spray drying installations: the EDECAD project. Drying Technology 22 (2004), 1403–1461 [22] J. Voigt, On substochastic C semigroups and their generators. In: Proceedings of the Conference on Mathematical Methods Applied to Kinetic Equations (Paris, 1985), Transport Theory Statist. Phys., no. 16, pp. 453–466 (1987) [23] J. Wells, Modelling coagulation in industrial spray drying: an efﬁcient one-dimensional population balance approach. Ph.D. thesis, University of Strathclyde, Department of Mathematics and Statistics (2018) [24] R. M. Ziff, Kinetics of polymerization. J. Statist. Phys. 23 (1980), 241–263 [25] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerisation. J. Phys. A 18 (1985), 3027–3037 Lyndsay Kerr, Wilson Lamb and Matthias Langer Department of Mathematics and Statistics University of Strathclyde 26 Richmond Street Glasgow G1 1XH UK E-mail: m.langer@strath.ac.uk Lyndsay Kerr E-mail: lyndsay.kerr@strath.ac.uk Wilson Lamb E-mail: w.lamb@strath.ac.uk http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals http://www.deepdyve.com/lp/springer-journals/discrete-fragmentation-systems-in-weighted-1-documentclass-12pt-Pcro1Qdq5R

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Publisher: Springer Journals
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ISSN: 1424-3199
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DOI: 10.1007/s00028-020-00561-6
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Abstract

J. Evol. Equ. Journal of Evolution © 2020 The Author(s) Equations https://doi.org/10.1007/s00028-020-00561-6 1 1 1 Discrete fragmentation systems in weighted spaces Lyndsay Kerr , Wilson Lamb and Matthias Langer Abstract. We investigate an inﬁnite, linear system of ordinary differential equations that models the evolu- tion of fragmenting clusters. We assume that each cluster is composed of identical units (monomers), and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Addi- tionally, we establish that it is always possible to identify a weighted space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions and show that, under appropriate restrictions on the fragmentation coefﬁcients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly deﬁned exponential rate. 1. Introduction There are many diverse situations arising in nature and industrial processes where clusters of particles can merge together (coagulate) to produce larger clusters and can break apart (fragment) to produce smaller clusters. Particular examples can be found in polymer science [1,24,25], in the formation of aerosols [16] and in the powder production industry [21,23]. It is often appropriate when modelling such processes to regard cluster size as a discrete variable, with a cluster of size n,an n-mer, composed of n identical units (monomers). By scaling the mass, we can assume that each monomer has unit mass and so an n-mer has mass n. The aim is to use the mathematical model to obtain information on how clusters of different sizes evolve. In this paper, we restrict our attention to the case when no coagulation occurs, and consequently the evolution of clusters can be described by a linear, inﬁnite system of ordinary differential equations. With the number density of clusters of size n (i.e. mass n) at time t denoted by u (t ), this fragmentation system is given by Mathematics Subject Classiﬁcation: 47D06, 34G10, 80A30, 34D05 Keywords: Discrete fragmentation, Positive semigroup, Analytic semigroup, Long-time behaviour, Sobolev towers. L. Kerr et al. J. Evol. Equ. u (t ) =−a u (t ) + a b u (t ), t > 0; n n j n, j j (1.1) j =n+1 u (0) = u˚ , n = 1, 2,..., n n where a is the rate at which clusters of size n are lost, b is the rate at which clusters n n, j of size n are produced when a larger cluster of size j fragments and u˚ is the initial density of clusters of size n at time t = 0. Equation (1.1) was ﬁrst introduced in [25]to deal with the case of binary fragmentation, where it is assumed that each fragmentation event results in the creation of exactly two daughter clusters. As in [7,10,18,19], we consider the more general case, where each fragmentation event can result in the creation of two or more clusters. Since (1.1) is an inﬁnite system, it is convenient to express solutions as time-dependent sequences of the form u(t ) := (u (t )) . n=1 Throughout this paper, we need various assumptions on the fragmentation coefﬁ- cients a and b . We list these assumptions here and will refer to them in the sequel n n, j when required. Assumption 1.1. (i) For all n ∈ N, a ≥ 0. (1.2) (ii) For all n, j ∈ N, b ≥ 0 and b = 0 when n ≥ j. (1.3) n, j n, j The total mass of daughter clusters resulting from the fragmentation of a j-mer j −1 is given by nb . In most papers that have dealt with discrete fragmentation n, j n=1 systems, it is assumed that j −1 nb ≤ j for all j = 2, 3,..., (1.4) n, j n=1 i.e. there is no increase in mass at fragmentation events. If there is strict inequality in (1.4), then mass is lost by some other mechanism. However, for most of our results we do not assume that (1.4) holds; this means that mass could even be gained at fragmen- tation events. We can specify the local mass loss or mass gain with real parameters λ , j = 2, 3,..., such that j −1 nb = (1 − λ ) j, j = 2, 3,.... (1.5) n, j j n=1 In terms of the densities u (t ), the total mass of all clusters in the system at time t is given by the ﬁrst moment, M (u(t )),of u(t ), where M u(t ) := nu (t ). (1.6) 1 n n=1 1 Discrete fragmentation systems in weighted spaces A formal calculation establishes that if u is a solution of (1.1), then M u(t ) =−a u (t ) − j λ a u (t ). (1.7) 1 1 1 j j j dt j =2 The expression in (1.7) gives the rate at which mass may be lost from the system or gained and also shows that, at least formally, the total mass is conserved when a = 0 and λ = 0 for all j = 2, 3,..., i.e. when j −1 a = 0 and nb = j for all j = 2, 3,.... (1.8) 1 n, j n=1 Note that monomers cannot fragment to produce smaller clusters, and hence the case when a > 0 is interpreted as a situation in which monomers are removed from the system. In this paper, the approach we use to investigate (1.1) relies on the theory of semi- groups of bounded linear operators and entails formulating (1.1) as an abstract Cauchy problem (ACP) in an appropriate Banach space. The existence and uniqueness of solu- tions to the ACP are established via the application of perturbation results for operator semigroups. Of particular relevance is the Kato–Voigt perturbation theorem for sub- stochastic semigroups [5,22] that was ﬁrst applied to (1.1)in[18], and subsequently in similar semigroup-based investigations into (1.1), such as [8,19]. We use a reﬁned version of this theorem proved by Thieme and Voigt in [20]. In previous studies, including [18,19], the ACP associated with the fragmentation system has been formulated in the space X := f = ( f ) : f ∈ R for all n ∈ N and n| f | < ∞ . (1.9) [1] n n n n=1 n=1 Equipped with the norm f = n| f |, f ∈ X , (1.10) [1] n [1] n=1 X is a Banach space, and [1] f = M ( f ) (1.11) [1] 1 if f ∈ X is such that f ≥ 0, n ∈ N. This means that whenever u :[0, ∞) → X [1] n [1] is a non-negative solution of the fragmentation system, the norm, u(t ) ,gives the [1] total mass at time t. Other Banach spaces, with norms related to higher-order moments, have also played a prominent role [8,11], with X being replaced by X , p > 1, [1] [ p] where ∞ p X := f = ( f ) : f ∈ R for all n ∈ N and f := n | f | < ∞ . [ p] n n [ p] n n=1 n=1 (1.12) L. Kerr et al. J. Evol. Equ. Rather than restricting our investigations to spaces of the type X , we choose to [ p] work within the framework of more general weighted spaces. As we shall demon- strate, this additional ﬂexibility will enable us to establish desirable semigroup prop- erties and results that may not always be possible in an X setting. Therefore, we let [ p] w = (w ) be such that w > 0 for all n ∈ N, and deﬁne n n n=1 1 ∞ = f = ( f ) : f ∈ R for all n ∈ N and w | f | < ∞ . (1.13) n n n n n=1 n=1 Equipped with the norm f = w | f |, f ∈ , (1.14) w n n n=1 1 1 is a Banach space, which we refer to as the weighted space with weight w. Motivated by the terms in (1.1), we introduce the formal expressions ⎛ ⎞ ∞ ∞ ∞ ⎝ ⎠ A : ( f ) → (−a f ) and B : ( f ) → a b f . n n n n j n, j j n=1 n=1 n=1 j =n+1 n=1 (w) (w) 1 Operator realisations, A and B ,of A and B, respectively, are deﬁned in by (w) (w) 1 1 A f = A f, D(A ) = f ∈ : A f ∈ (1.15) w w and (w) (w) 1 1 B f = B f, D(B ) = f ∈ : B f ∈ . (1.16) w w Here, and in the sequel, D(T ) denotes the domain of the designated operator T . −1 Similarly, we shall represent the resolvent, (λI − T ) ,of T by R(λ, T ). An ACP version of (1.1), posed in the space , can be formulated as (w) (w) u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚. (1.17) Note that this reformulation of (1.1) imposes additional constraints on both the initial data and the sought solutions since we now require u˚ ∈ and also that the (w) (w) solution u(t ) ∈ D(A ) ∩ D(B ) for all t > 0. Moreover, as the derivative on the left-hand side of (1.17) is deﬁned in terms of · , it is customary to look for 1 1 1 a solution u ∈ C ((0, ∞), ) ∩ C ([0, ∞), ). Such a solution is referred to as a w w classical solution of (1.17) and has the property that u(t ) − u˚ → 0as t → 0 . (w) (w) It turns out that often, instead of using the operator A + B on the right-hand side of (1.17), one has to use its closure, which leads to the ACP (w) (w) u (t ) = (A + B )u(t ), t > 0; u(0) = u˚. (1.18) Yet another option for an operator on the right-hand side is the maximal operator, (w) G , which is deﬁned by max (w) (w) 1 1 G f = A f + B f, D G = f ∈ : A f + B f ∈ . (1.19) max max w w 1 Discrete fragmentation systems in weighted spaces However, the domain of this operator is too large in general to ensure uniqueness of solutions; see Example 4.3 and also [6] where a continuous fragmentation equation is studied. There are a number of beneﬁts to be gained by working in more general weighted spaces, least of which is the derivation of existence and uniqueness results for (1.1) in that reduce to those established in earlier X -based investigations by choosing [ p] p (w) (w) (w) w = n . For example, in Theorem 3.4 we prove that G = A + B is the generator of a substochastic C -semigroup. While this result has already been shown for the speciﬁc case w = n for p ≥ 1, see [8,18], Theorem 3.4 is formulated for more general weights and is proved by means of an alternative and novel argument that is based on theory presented in [20]. Our approach also leads to an additional invariance result, which can be used to establish the existence of solutions to the fragmentation system (1.17) for a certain speciﬁed class of initial conditions. A further major advantage of working in the more general setting of is that it yields results on the analyticity of the related fragmentation semigroups, which do not necessarily hold in the restricted case of w = n , p ≥ 1. In particular, in Theorem 5.5 we prove that, for any fragmentation coefﬁcients, we can always (w) (w) ﬁnd a weight w such that A + B is the generator of an analytic, substochastic C -semigroup on . In connection with this, it should be noted that there are no known general results that guarantee the analyticity of the fragmentation semigroup on the space X . Indeed, this provided the motivation for previous investigations into [1] fragmentation ACPs posed in higher moment spaces, which led to a sufﬁcient condition (w) (w) being found in [8]for A + B to generate an analytic semigroup on X for some [ p] p > 1. However, simple examples are also given in [8] of fragmentation coefﬁcients where the semigroup is not analytic in X for any p ≥ 1; see Example 5.6. [ p] The importance of establishing the analyticity of the semigroup associated with the fragmentation system is that analytic semigroups have extremely useful properties. (w) (w) 1 For example, if A + B generates an analytic semigroup on , then it follows immediately that the ACP (1.17) has a unique classical solution for any u˚ ∈ . In addition, when coagulation is introduced into the system, the analyticity of the (w) (w) semigroup generated by A + B can be used to weaken the assumptions that are required on the cluster coagulation rates to obtain the existence and uniqueness of solutions to the corresponding coagulation–fragmentation system of equations. Such coagulation–fragmentation systems will be considered in a subsequent publication. Once the well-posedness of the fragmentation ACP has been satisfactorily dealt with, the next question to be addressed is that of the long-term behaviour of solutions. Results on the asymptotic behaviour of solutions to (1.17)are givenin[7,12,15]for the speciﬁc case where the weight is w = n for p ≥ 1, n ∈ N. In particular, for mass-conserving fragmentation processes, where (1.8) holds, it is shown that the solution of (1.17) converges to a state where there are only monomers present if and only if a > 0 for all n ≥ 2. In Sect. 6, we continue to work with more general weights and, in the mass loss case, show that the solution of (1.17) decays to the zero state L. Kerr et al. J. Evol. Equ. over time if and only if a > 0 for all n ∈ N. This mass loss result can then be used to deduce that the solution, in the mass-conserving case, converges to the monomer state if and only if a > 0 for all n ≥ 2, this result now holding in the general weighted space . Regarding the rate at which solutions approach the steady state, the case where mass is conserved and w = n for p > 1 is examined in [12, Section 4], and it is shown that solutions decay to the monomer state at an exponential rate, which, however, is not quantiﬁed. In Sect. 6, we obtain results regarding the exponential rate of decay of solutions, both for the mass-conserving and for the mass loss cases, by working in a 1 (w) (w) space in which A + B generates an analytic semigroup. The approach we use enables us to quantify the exponential decay rate. In [19], the theory of Sobolev towers is used to investigate a speciﬁc example of (1.1) that has been proposed as a model of random bond annihilation. Of particular note is the fact that the resulting analysis provides a rigorous explanation of an apparent non- uniqueness of solutions that emanate from a zero initial condition. We shall establish that an approach involving Sobolev towers can also be used to obtain results on (1.1) , where w is for general fragmentation coefﬁcients. By writing (1.1)asanACP in (w) (w) 1 such that A + B generates an analytic, substochastic C -semigroup on ,we are able to construct a Sobolev tower and then use this to prove the existence of unique, non-negative solutions of (1.17) for a wider class of non-negative initial conditions than those in ; see Theorem 7.2. The paper is structured as follows. In Sect. 2, we provide some prerequisite results and deﬁnitions. Following this, we begin our examination of (1.1) in Sect. 3, obtaining, in particular, the aforementioned Theorem 3.4, which is then used to draw conclusions on the existence and uniqueness of solutions to (1.17) and (1.18), both in the space X and in more general spaces. We consider the pointwise system (1.1) in Sect. 4 [1] and show that for any u˚ ∈ , a solution of (1.1) can be expressed in terms of the (w) (w) (w) semigroup generated by G = A + B . We then use this result to show that (w) (w) G is a restriction of the maximal operator G . This is important in investigations max into the full coagulation–fragmentation system as it allows the fragmentation terms (w) to be completely described by the operator G . Results on the analyticity of the fragmentation semigroup are presented in Sect. 5 and then applied both in Sect. 6, where the asymptotic behaviour of solutions is investigated, and in Sect. 7, where the theory of Sobolev towers is applied to establish the well-posedness of (1.17)for more general initial conditions. 2. Preliminaries We begin by recalling some terminology. The following notions are well known and can be found in various sources, including [9,13]. Let X be a real vector lattice with norm ·. The positive cone, X ,of X is the set of non-negative elements in X and, similarly, for a subspace D of X, we denote the set of non-negative elements in D 1 Discrete fragmentation systems in weighted spaces by D .If X is a vector lattice, then for each f ∈ X the vectors f := sup{± f, 0} are + ± well deﬁned and satisfy f , f ∈ X and f = f − f . A vector lattice, equipped + − + + − with a lattice norm ·,issaidtobea Banach lattice if X is complete under ·. Moreover, if the lattice norm satisﬁes f + g= f +g for all f, g ∈ X , then X is an AL-space. It can be shown that, when X is an AL-space, there exists a unique, bounded linear functional, φ, that extends · from X to X; see [9, Theorems 2.64 and 2.65]. We now turn our attention to C -semigroups which are crucial to our investigation into the pure fragmentation system. The notions and results given here can be found in [17]. First, we note that if (S(t )) is a C -semigroup on a Banach space X, then t ≥0 0 ωt there exist M ≥ 1 and ω ∈ R such that S(t )≤ Me for all t ≥ 0, and the growth bound, ω ,of (S(t )) is deﬁned by 0 t ≥0 ωt ω := inf ω ∈ R : there exists M ≥ 1 such that S(t )≤ M e for all t ≥ 0 . 0 ω ω Analytic semigroups, see [17, Deﬁnition II.4.5], are of particular importance in Sect. 5. Semigroups of this type have a number of useful properties that make them desirable to work with. For example, if G is the generator of an analytic semigroup, (S(t )) , on a Banach space X, then S(t ) f ∈ D(G ) for all t > 0, n ∈ N and f ∈ X, t ≥0 and S(·) is inﬁnitely differentiable. When dealing with many physical problems, such as the fragmentation system, meaningful solutions must be non-negative, and this requirement has to be taken into account in any semigroup-based investigation. In connection with this, we say that a C -semigroup (S(t )) on an ordered Banach space X, such as a Banach lattice, is 0 t ≥0 positive if S(t ) f ≥ 0 for all f ∈ X ;itiscalled substochastic (resp. stochastic)if, additionally, S(t ) f ≤ f (resp. S(t ) f = f ) for all f ∈ X . It follows that if G generates a substochastic semigroup (S(t )) , then the associated ACP t ≥0 u (t ) = Gu(t ), t > 0; u(0) = u˚, has a unique, non-negative classical solution, given by u(t ) = S(t )u˚, for any u˚ ∈ D(G) . A result on substochastic semigroups and their generators that we shall exploit is due to Thieme and Voigt [20, Theorem 2.7]. This result gives sufﬁcient conditions (w) (w) under which the closure of the sum of two operators, such as A + B in (1.17), generates a substochastic semigroup. The existence of an invariant subspace under the resulting semigroup is also established. As we demonstrate in Proposition 2.4,itis possible to adapt the Thieme–Voigt result to produce a modiﬁed version that is ideally suited for applying to the fragmentation system. We ﬁrst provide some prerequisite results that are used in the proof of this proposition. L. Kerr et al. J. Evol. Equ. Lemma 2.1. Let A be a closable operator in a Banach space X. If G = Ais the generator of a C -semigroup on X, then no other extension of A is the generator of a C -semigroup on X. Proof. Suppose that G = A and H ⊇ A are generators of C -semigroups with growth bounds ω and ω , respectively, and assume that H = G. Clearly, H ⊇ G since H is 1 2 closed. Let λ> max{ω ,ω }. Then, λ ∈ ρ(G)∩ρ(H ) and hence λI −G : D(G) → X 1 2 and λI − H : D(H ) → X are both bijective. This is a contradiction since λI − H is a proper extension of λI − G. The following lemma, which is a special case of [9, Remark 6.6], will also be used. For the convenience of the reader, we present a short proof. Lemma 2.2. Let G be the generator of a positive C -semigroup on a Banach lattice X. Then, for every f ∈ D(G), there exist g, h ∈ D(G) such that f = g − h. Proof. Let f ∈ D(G). Further, let ω be the growth bound of the semigroup generated by G,ﬁx λ>ω and set f := (λI − G) f . Since X is a Banach lattice, we have 0 0 f = f − f with f , f ∈ X .Now,let g := R(λ, G) f and h := R(λ, G) f . 0 + − + − + + − The fact that G generates a positive semigroup implies that R(λ, G) is a positive operator, and therefore g, h ∈ D(G) . Moreover, f = R(λ, G) f = R(λ, G)( f − f ) = R(λ, G) f − R(λ, G) f = g − h, 0 + − + − which proves the result. When the fragmentation coefﬁcients satisfy Assumption 1.1 and (1.8), then, as mentioned in the previous section, a formal calculation shows that the total mass is conserved. Consequently, if u is a non-negative solution of the fragmentation system, and it is known that u(t ) ∈ X for t ≥ 0, then we would expect u to satisfy [1] ∞ ∞ u(t ) = nu (t ) = nu˚ =u˚ for all t ≥ 0. [1] n [1] n=1 n=1 Clearly, this mass conservation property will hold whenever the solution can be written in terms of a stochastic semigroup on X . To this end, the following proposition will [1] prove useful. Proposition 2.3. Let (S(t )) be a positive C -semigroup on an AL-space, X, with t ≥0 0 generator G, and let φ be the unique bounded linear extension of the norm · from X to X. (i) The semigroup (S(t )) is stochastic if and only if t ≥0 φ S(t ) f = φ( f ) for all f ∈ X. (2.1) (ii) If φ(Gf ) = 0 for all f ∈ D(G) , then (2.1) holds and hence the semigroup (S(t )) is stochastic. t ≥0 1 Discrete fragmentation systems in weighted spaces (iii) Let G be an operator such that G = G .If φ(G f ) = 0 for all f ∈ D(G ) 0 0 0 0 + and each f ∈ D(G ) can be written as f = g − h, where g, h ∈ D(G ) , then 0 0 + (2.1) holds and hence (S(t )) is stochastic. t ≥0 Proof. (i) Assume that (S(t )) is stochastic and let f ∈ X and t ≥ 0. Then, t ≥0 f = f − f , where f , f ∈ X , and therefore + − + − + φ S(t ) f = φ S(t ) f − φ S(t ) f =S(t ) f −S(t ) f + − + − = f − f = φ( f ) − φ( f ) = φ( f ). + − + − Conversely, when (2.1) holds, we have S(t ) f = φ(S(t ) f ) = φ( f ) = f for f ∈ X and t ≥ 0. (ii) Let f ∈ D(G). From Lemma 2.2, there exist g, h ∈ D(G) such that f = g −h. Then, d d φ(S(t ) f ) = φ S(t ) f = φ GS(t ) f dt dt = φ GS(t )g − φ GS(t )h = 0 since S(t )g, S(t )h ∈ D(G) . Thus, φ(S(t ) f ) = φ( f ) for all f ∈ D(G), and hence also for all f ∈ X, since D(G) is dense in X. (iii) Let f ∈ D(G ). Then, f = g − h for some g, h ∈ D(G ) by assumption, and 0 0 + φ(G f ) = φ G (g − h) = φ(G g) − φ(G h) = 0. 0 0 0 0 Thus, φ(G f ) = 0 for all f ∈ D(G ).Now,let f ∈ D(G). Then, there exist 0 0 (n) (n) (n) f ∈ D(G ), n ∈ N, such that f → f and G f → Gf as n →∞. 0 0 Therefore, (n) (n) φ(Gf ) = φ lim G f = lim φ(G f ) = 0, 0 0 n→∞ n→∞ and the result follows from part (ii). We now use [20, Theorem 2.7] to obtain the following proposition, which will later be applied to the fragmentation problem. Proposition 2.4. Let (X, ·) and (Z , · ) be AL-spaces, such that (i) Z is dense in X, (ii) (Z , · ) is continuously embedded in (X, ·). Also, let φ and φ be the linear extensions of · from X to X and of · from Z + Z Z to Z, respectively. Let A : D(A) → X, B : D(B) → X be operators in X such that D(A) ⊆ D(B). Assume that the following conditions are satisﬁed. (a) −A is positive; (b) A generates a positive C -semigroup, (T (t )) ,on X; 0 t ≥0 L. Kerr et al. J. Evol. Equ. (c) the semigroup (T (t )) leaves Z invariant and its restriction to Z is a (neces- t ≥0 sarily positive) C -semigroup on (Z , · ), with generator A given by 0 Z Af = A f for all f ∈ D(A) = f ∈ D(A) ∩ Z : Af ∈ Z ; (d) B| is a positive linear operator; D(A) (e) φ((A + B) f ) ≤ 0 for all f ∈ D(A) ; (f) (A + B) f ∈ Z and φ ((A + B) f ) ≤ 0 for all f ∈ D(A) ; Z + (g) Af ≤ f for all f ∈ D(A) . Z + Then, there exists a unique substochastic C -semigroup on X which is generated by an extension, G, of A + B. The operator G is the closure of A + B. Moreover, the semigroup (S(t )) generated by G leaves Z invariant. If φ((A + B) f ) = 0 for all t ≥0 f ∈ D(A) , then (S(t )) is stochastic. + t ≥0 Proof. We ﬁrst show that the conditions of [20, Theorem 2.7] hold. From (ii) and the fact that (Z , · ) is an AL-space, it is clear that [20, Assumption 2.5] is satisﬁed. Also, from (f) and (g) we obtain that φ (A + B) f ≤ 0 ≤ f −Af Z Z for all f ∈ D(A) . Moreover, (f) and the deﬁnition of A imply that Bf ∈ Z for all f ∈ D(A) . Consequently, if we now take f ∈ D(A) and use Lemma 2.2 to express Bf as Bg − Bh, where g, h ∈ D(A) , then it follows easily that B(D(A)) ⊆ Z. Thus, all the assumptions of [20, Theorem 2.7] are satisﬁed and therefore G = A + B is the generator of a substochastic semigroup (S(t )) , which leaves Z invariant. t ≥0 That no other extension of A + B can generate a C -semigroup on X is an immediate consequence of Lemma 2.1. Finally, since A generates a substochastic C -semigroup, it follows from Lemma 2.2 that we can write any f ∈ D(A) = D(A + B) as f = g − h, where g, h ∈ D(A) . An application of Proposition 2.3 (iii) then yields the stochasticity result. 3. The fragmentation semigroup In this section, we begin our analysis of the fragmentation system (1.1) by investi- gating the associated ACP (1.17), which we recall takes the form (w) (w) u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚, (w) (w) 1 where A and B are deﬁned in by (1.15) and (1.16), respectively. A direct application of Proposition 2.4 will establish that, under appropriate conditions on the (w) (w) (w) (w) weight w, G = A + B generates a substochastic C -semigroup, (S (t )) , 0 t ≥0 1 (w) (w) 1 on . As no other extension of A + B generates a C -semigroup on ,we w w (w) 1 shall refer to (S (t )) as the fragmentation semigroup on . In the process of t ≥0 proving the existence of the fragmentation semigroup, we shall also obtain explicit 1 (w) subspaces of which are invariant under (S (t )) . t ≥0 w 1 Discrete fragmentation systems in weighted spaces First, we note that is an AL-space, with positive cone 1 ∞ 1 = f = ( f ) ∈ : f ≥ 0 for all n ∈ N , n n w n=1 w whenever w = (w ) is a positive sequence. Moreover, in this case the unique n=1 1 1 bounded linear functional, φ , that extends · from ( ) to is given by w w + w w φ ( f ) = w f for all f ∈ . (3.1) w n n n=1 We recall also that if we take w = n for all n ∈ N, then = X and · = · . n [1] w [1] (w) (w) For this speciﬁc case, we shall represent φ , A and B by M , A and B , w 1 1 1 respectively, and consequently the ACP (1.17)on X will be written as [1] u (t ) = A u(t ) + B u(t ), t > 0; u(0) = u˚. (3.2) 1 1 From physical considerations, it is clear that the initial condition, u˚,inthe ACP (1.17) must necessarily be non-negative, and similarly, if u :[0, ∞) → is the cor- responding solution, then we require u(t ) to be non-negative for all t ≥ 0. Moreover, if we assume (1.4) to hold, or, equivalently, (1.5) with λ ∈[0, 1], we expect from (1.7) that mass is either lost or conserved during fragmentation. From (1.6) and the deﬁnition of the norm on X , this is equivalent to [1] u(t ) ≤u˚ for all t ≥ 0, (3.3) [1] [1] with equality being required in the mass-conserving case, provided that w is such that ⊆ X . [1] For convenience, we include the following elementary result which states that (w) 1 the operator A generates a substochastic semigroup on for any non-negative weight w. Lemma 3.1. Let and · be deﬁned by (1.13) and (1.14), respectively, and (w) let (1.2) hold. Then, the operator A , deﬁned by (1.15), is the generator of a sub- (w) 1 stochastic C -semigroup, (T (t )) ,on , which is given, for t ≥ 0, by the inﬁnite 0 t ≥0 −a t diagonal matrix diag(v (t ), v (t),...), where v (t ) = e for all n ∈ N. 1 2 n For the remainder of this section, the weight, w, will be required to satisfy the following assumption. Assumption 3.2. (i) w ≥ n for all n ∈ N. (ii) There exists κ ∈ (0, 1] such that j −1 w b ≤ κw for all j = 2, 3,.... (3.4) n n, j j n=1 L. Kerr et al. J. Evol. Equ. Remark 3.3. Let w be such that (w /n) is increasing and let (1.4) hold. Then, n=1 j −1 j −1 j −1 w w w j j w b = nb ≤ nb ≤ j = w . n n, j n, j n, j j n j j n=1 n=1 n=1 Hence, (3.4) is satisﬁed with κ = 1. In particular, if (1.4) holds, then Assumption 3.2 is automatically satisﬁed by any weight of the form w = n , p ≥ 1. (w) It is an immediate consequence of Assumption 3.2 that, for any f ∈ D(A ) ,we have ⎛ ⎞ j −1 ∞ ∞ ∞ (w) ⎝ ⎠ φ B f = w a b f = w b a f w n j n, j j n n, j j j n=1 j =n+1 j =2 n=1 (3.5) (w) ≤ κ w a f =−κφ A f . j j j w j =1 (w) Consequently, for all f ∈ D(A ), ∞ ∞ (w) (w) B f = w a b f ≤ φ B | f | w n j n, j j w (3.6) n=1 j =n+1 (w) (w) ≤−κφ A | f | = κA f , w w from which it follows that (w) (w) (w) (w) (w) (w) (w) D(A ) ⊆ D(B ) and D A + B = D(A ) ∩ D(B ) = D(A ). (3.7) (w) (w) We now apply Proposition 2.4 to the operators A and B . This involves the construction of a suitable subspace of , and to this end we require a sequence (c ) that satisﬁes n=1 c ≤ c and a ≤ c for all n ∈ N. (3.8) n n+1 n n Note that such a sequence can always be found. For example, we can take c = max{a ,..., a } for n = 1, 2,.... (3.9) n 1 n (w) Let C be the corresponding multiplication operator, deﬁned by (w) (w) 1 [C f ] =−c f , n ∈ N, D(C ) = f ∈ : w c | f | < ∞ , n n n n n n n=1 (3.10) (w) and equip D(C ) with the graph norm (w) (w) f (w) = f +C f = (w + w c )| f |, f ∈ D(C ). (3.11) w w n n n n n=1 1 Discrete fragmentation systems in weighted spaces (w) 1 ∞ Clearly, (D(C ), · (w) ) = ( , · ) with weight w = (w ) where w n w n=1 w = w + w c , n ∈ N, (3.12) n n n n and hence ( , · ) is an AL-space, and the unique linear extension of · from w w 1 1 1 ( ) to is given by φ ( f ) = w f for f ∈ . + w n n w w n=1 w ∞ ∞ We note that the choice (3.9)for (c ) is ‘maximal’ in the sense that if (cˆ ) n n n=1 n=1 is any other monotone increasing sequence that dominates (a ) and C is deﬁned n=1 (w) (w) analogously to (3.10), then D(C ) ⊆ D(C ). (w) (w) (w) Theorem 3.4. Let Assumptions 1.1 and 3.2 hold. Then, G = A + B is the (w) 1 generator of a substochastic C -semigroup, (S (t )) ,on . Moreover, the semi- 0 t ≥0 (w) (w) 1 (w) group (S (t )) leaves D(C ) = invariant, where D(C ) and w are deﬁned t ≥0 in (3.10) and (3.12), respectively, and (c ) satisﬁes (3.8). If, in addition, (1.8) holds n=1 and w = n for all n ∈ N, then the semigroup, (S (t )) , generated by G = A + B n 1 t ≥0 1 1 1 is stochastic on X . [1] Proof. We show that the conditions (i), (ii) and (a)–(g) of Proposition 2.4 are all (w) (w) satisﬁed when A = A , B = B and the AL-spaces (X, ·) and (Z , · ) are, 1 (w) 1 respectively, and (D(C ), · (w) ) = ( , · ). w C 1 1 Clearly, is dense in and continuously embedded since w ≤ w , n ∈ N.It n n follows that (i) and (ii) both hold. (w) Condition (a) is obviously satisﬁed by A , and, for (b), we apply Lemma 3.1 to (w) (w) 1 establish that A generates a substochastic C -semigroup, (T (t )) ,on .Itis 0 t ≥0 (w) 1 easy to see that the semigroup (T (t )) leaves invariant and the generator of t ≥0 1 (w) (w) 1 the restriction to is A , the part of A in ; this shows (c). w w (w) (w) It is also clear that B is positive. From (3.5), we obtain that, for f ∈ D(A ) , (w) (w) (w) (w) φ A + B f = φ (A f ) + φ (B f ) w w w (3.13) (w) (w) ≤ φ (A f ) − κφ (A f ) ≤ 0. w w Hence, (d) and (e) hold. Since w ≥ n, by Assumption 3.2 (i), we have w = w + w c ≥ n, n ∈ N. n n n n n Moreover, the monotonicity of (c ) and Assumption 3.2 (ii) imply that n=1 j −1 j −1 j −1 w b = (1 + c )w b ≤ (1 + c ) w b ≤ κ(1 + c )w = κw n n, j n n n, j j n n, j j j j n=1 n=1 n=1 for all j ∈ N. This means that Assumption 3.2 also holds for the weight w . Therefore, (w) (w) (w) (w) we obtain from (3.7) and (3.13) that D(A ) ⊆ D(B ) and φ ((A +B ) f ) ≤ 0 (w) for f ∈ D(A ) , and so (f) is also satisﬁed. That (g) holds follows from + L. Kerr et al. J. Evol. Equ. ∞ ∞ ∞ (w) A f = w a | f |≤ w c | f |≤ w | f |= f w n n n n n n n n w n=1 n=1 n=1 (w) for f ∈ D(A ) . (w) Thus, the conditions of Proposition 2.4 are all satisﬁed and therefore G = (w) (w) (w) 1 A + B is the generator of a substochastic C -semigroup, (S (t )) ,on , 0 t ≥0 (w) 1 which also leaves D(C ) = invariant. Finally, assume that (1.8) is satisﬁed and w = n for all n ∈ N. Then, equality holds in (3.4) with κ = 1 and hence also in (3.5), and so, from Proposition 2.4,the semigroup generated in this case is stochastic. Remark 3.5. Consider the case where w = n for all n ∈ N, so that = X , n [1] and let Assumption 1.1 and (1.4) hold. Then, by Remark 3.3,(3.4) is also satisﬁed, and therefore, from Theorem 3.4, the operator G = A + B is the generator of a 1 1 1 substochastic C -semigroup, (S (t )) ,on X . It follows that the ACP 0 1 t ≥0 [1] u (t ) = G u(t ), t > 0; u(0) = u˚, (3.14) with u˚ ∈ D(G ), has a unique classical solution, given by u(t ) = S (t )u˚ for all t ≥ 0. 1 1 Moreover, if u˚ ≥ 0, then this solution is non-negative. Now suppose that u˚ ∈ D(G ) 1 + and, in addition, assume that (1.8) holds. Then, the semigroup (S (t )) is stochastic 1 t ≥0 on X and so, from (1.11), [1] M u(t ) =u(t ) =S (t )u˚ =u˚ = M (u˚) for all t ≥ 0, 1 [1] 1 [1] [1] 1 showing that u(t ) is a mass-conserving solution. With the help of Remark 3.5, we obtain the following corollary. (w) (w) Corollary 3.6. Let Assumptions 1.1 and 3.2 hold and let u˚ ∈ D(G ), where G = (w) (w) A + B as in Theorem 3.4. Then, the ACP (w) u (t ) = G u(t ), t > 0; u(0) = u˚ (3.15) (w) has a unique classical solution, given by u(t ) = S (t )u˚. This solution is non-negative (w) (w) if u˚ ∈ D(G ) . Moreover, if (1.8) holds and u˚ ∈ D(G ) , then this solution is + + mass conserving. (w) Proof. It follows immediately from Theorem 3.4 that u(t ) = S (t )u˚ is the unique (w) (w) classical solution of (3.15) for all u˚ ∈ D(G ). Moreover, since (S (t )) is t ≥0 (w) substochastic, this solution is non-negative if u˚ ∈ D(G ) . (w) Now, assume that (1.8) holds and u˚ ∈ D(G ) . Then, (S (t )) is a stochastic + 1 t ≥0 C -semigroup on X . Additionally, since w ≥ n for all n ∈ N, is continuously 0 [1] n embedded in X and so, as u(t ) is differentiable in , u(t ) is also differentiable in [1] (w) 1 X and the derivatives must coincide. Moreover, since G is the part of G in , [1] 1 (w) we have u(t ) ∈ D(G ). Therefore, u(t ) = S (t )u˚ is also a solution of (3.14), and, (w) by uniqueness of solutions, it follows that S (t )u˚ = S (t )u˚ for t ≥ 0. Remark 3.5 (w) then establishes that u(t ) = S (t )u˚ is a mass-conserving solution. 1 Discrete fragmentation systems in weighted spaces (w) Note that even if u ∈ D(A ), the solution, u(t ),of (3.15) need not belong to (w) D(A ) for any t > 0. Hence, the existence of a solution of (1.17) is not guaranteed in general; one only has uniqueness of solutions. However, the next theorem shows (w) that under the stronger assumption u˚ ∈ D(C ) on the initial condition, the ACP (1.17) is well posed. (w) Theorem 3.7. Let Assumptions 1.1 and 3.2 hold. For u˚ ∈ D(C ),the ACP (1.17) (w) (w) has a unique classical solution given by u(t ) = S (t )u, ˚ t ≥ 0.If u˚ ∈ D(C ) , (w) then this solution is non-negative. Moreover, if (1.8) holds and u˚ ∈ D(C ) , then the solution is mass conserving. (w) (w) (w) (w) Proof. We know that G and A + B coincide on D(A ) and also that (w) (w) (w) u(t ) = S (t )u˚ is the unique solution of (3.15)for u˚ ∈ D(C ) ⊆ D(G ). Since (w) (w) (w) (w) (w) (S (t )) leaves D(C ) invariant, it follows that S (t )u˚ ∈ D(C ) ⊆ D(A ). t ≥0 The result then follows from Corollary 3.6. The next proposition shows that if the sequence (a ) has a certain additional n=1 (w) property, then a unique solution of (1.17)existsfor u˚ ∈ D(A ). Proposition 3.8. Let (a ) be an unbounded sequence such that (1.2) holds. Fur- n=1 ∞ ∞ ther, deﬁne the sequence (c ) by (3.9) and let w = (w ) be such that w > 0 n n n n=1 n=1 (w) (w) for all n ∈ N. Then, D(C ) = D(A ) if and only if lim inf > 0. (3.16) n→∞ Proof. Note ﬁrst that the unboundedness of (a ) implies that c →∞ as n →∞. n n n=1 (w) (w) Since c ≥ a for all n ∈ N,wehave D(C ) ⊆ D(A ).If (3.16) holds, then there n n (w) exist γ> 0, N ∈ N such that a ≥ γ c for all n ≥ N.Let f ∈ D(A ). Then, n n ∞ N −1 ∞ (w) C f = w c | f |≤ w c | f |+ w a | f | w n n n n n n n n n n=1 n=1 n=N N −1 (w) ≤ w c | f |+ A f < ∞, n n n w n=1 (w) (w) and so D(A ) = D(C ). Now, suppose that lim inf (a /c ) = 0. Then, there exists a subsequence, n→∞ n n a /c , such that n n k k k=1 a 1 1 1 c = 0, ≤ and ≤ for all k ∈ N. c k c k n n k k Let f be such that 1/(c w k) when j = n , n n k k k f = (3.17) 0 otherwise. L. Kerr et al. J. Evol. Equ. Then, ∞ ∞ ∞ 1 1 a w | f |= a w ≤ < ∞, n n n n n k k c w k k n n k k n=1 k=1 k=1 ∞ ∞ ∞ 1 1 w | f |= w ≤ < ∞, n n n c w k k n n k k n=1 k=1 k=1 ∞ ∞ c w | f |= =∞. n n n n=1 k=1 (w) (w) (w) It follows that f ∈ D(A )\D(C ), showing that D(C ) is a proper subset of (w) D(A ). Remark 3.9. If (a ) is unbounded and eventually monotone increasing, then n=1 ∞ (w) (c ) , given by (3.9), satisﬁes (3.16). Note that, in X , the invariance of D(A ) n [1] n=1 under the fragmentation semigroup has already been established in [18, Theorem 3.2] for the case when (a ) is monotone increasing. n=1 We end this section by obtaining an inﬁnite matrix representation of the fragmen- (w) 1 tation semigroup (S (t )) on , which is used in Sect. 6. Let Assumptions 1.1 t ≥0 (w) (w) (w) and 3.2 be satisﬁed so that G = A + B is the generator of a substochastic (w) 1 1 C -semigroup, (S (t )) ,on .For n ∈ N,let e ∈ be given by 0 t ≥0 n w w 1if n = k, (e ) = (3.18) n k 0 otherwise, and let (s (t )) be the inﬁnite matrix deﬁned by m,n m,n∈N (w) s (t ) = (S (t )e ) for all m, n ∈ N. m,n n m (w) Note that, since (S (t )) is positive, s (t ) ≥ 0 for all m, n ∈ N. Now, each t ≥0 m,n f ∈ can be expressed as f = f e , where the inﬁnite series is convergent n n w n=1 in . Hence, ∞ ∞ (w) (w) S (t ) f = f S (t )e = f s (t ) for all m ∈ N, n n n m,n n=1 n=1 (w) and therefore (S (t )) can be represented by the matrix (s (t )) . To deter- t ≥0 m,n m,n∈N mine s (t ) more explicitly, ﬁx n ∈ N and let (u (t), ..., u (t )) be the unique m,n 1 n solution of the n-dimensional system u (t ) =−a u (t ) + a b u (t ), t > 0; m = 1, 2,..., n; (3.19) m m j m, j j j =m+1 u (0) = 1; u (0) =0for m < n. (3.20) n m 1 Discrete fragmentation systems in weighted spaces It is straightforward to check that u(t ) = (u (t ), . . . , u (t ), 0, 0,...) solves (1.1) 1 n (w) (w) with u˚ = e . Since u(t ) ∈ D(A ) ⊆ D(G ), the function u coincides with the (w) unique solution of (3.15), and hence u(t ) = S (t )e , which yields u (t ), m = 1, 2,..., n, s (t ) = (3.21) m,n 0, m > n. For m = n, the differential equation in (3.19) reduces to u (t ) =−a u (t ), which n n −a t implies that s (t ) = u (t ) = e . Since n was arbitrary, it follows that, for all n,n n t ≥ 0, ⎡ ⎤ −a t e s (t ) s (t ) ··· 1,2 1,3 ⎢ ⎥ ⎡ ⎤ (w) −a t ⎢ ⎥ e S (t ) −a t ⎢ ⎥ (12) 0 e s (t ) ··· 2,3 (w) ⎢ ⎥ ⎣ ⎦ S (t ) = = , (3.22) ⎢ ⎥ −a t (w) ⎢ ⎥ 0 0 e ··· 0 S (t ) (22) ⎣ ⎦ . . . . . . . . . . (w) where 0 is an inﬁnite column vector consisting entirely of zeros, S (t ) is a non- (12) (w) negative inﬁnite row vector and S (t ) is an inﬁnite-dimensional, non-negative, upper (22) triangular matrix. We note that, in the particular case when = X and mass is [1] conserved, Banasiak obtains the inﬁnite matrix representation (3.22) for the semigroup (S (t )) in [7, Equation (10) and Lemma 1]. In [7], an explicit expression is also 1 t ≥0 found for s (t ), m < n, but we omit this here since it is not required for the results m,n that follow. As observed in [7, pp. 363], it follows from (3.22) that, for all N ∈ N,we have S(t ) f ∈ span{e , e ,..., e } for all f ∈ span{e , e ,..., e }. Also, note that 1 2 N 1 2 N the functions s are independent of the weight w, which implies that, whenever w is m,n (w) (w) 1 1 another weight satisfying Assumption 3.2, S (t ) and S (t ) coincide on ∩ . w w 4. The pointwise fragmentation problem and the fragmentation generator We established in Theorem 3.7 that if Assumptions 1.1 and 3.2 are satisﬁed, then (w) u(t ) = S (t )u˚ is the unique, non-negative classical solution of the fragmentation (w) ACP (1.17) for all u˚ ∈ D(C ) . Moreover, when (1.8) holds, then this solution is (w) mass conserving. Clearly, u(t ) = S (t )u˚ will also satisfy the fragmentation system (w) (1.1) in a pointwise manner when u˚ ∈ D(C ) . However, at this stage we do (w) not know in what sense, if any, the semigroup (S (t )) provides a non-negative t ≥0 solution for a general u˚ ∈ ( ) . In this section, we show that a non-negative solution of the pointwise system (1.1) can be determined for any given initial condition in 1 (w) ( ) by using the semigroup (S (t )) . + t ≥0 As before, we require Assumptions 1.1 and 3.2 to hold, and we deﬁne a sequence ∞ (w) (c ) by (3.9), with the associated multiplication operator C given by (3.10). n=1 L. Kerr et al. J. Evol. Equ. (w) (w) Then, a ≤ c for all n ∈ N and it follows that D(C ) ⊆ D(A ). From Propo- n n (w) (w) sition 3.4, D(C ) is invariant under the substochastic semigroup (S (t )) gen- t ≥0 (w) (w) (w) (w) erated by G = A + B . Consequently, u(t ) = S (t )u˚ is the unique, non- (w) negative classical solution of (1.17) for each u˚ ∈ D(C ) , and therefore t t u (t ) − u˚ =−a u (s) ds + a b u (s) ds, (4.1) n n n n j n, j j 0 0 j =n+1 for n = 1, 2,.... We use this integrated version of the pointwise fragmentation system (1.1) to prove the following result. Theorem 4.1. Let Assumptions 1.1 and 3.2 hold, and let u˚ ∈ . Then, u(t ) = (w) S (t )u˚ satisﬁes the system (1.1) for almost all t ≥ 0. Moreover, if u˚ ≥ 0, then u(t ) ≥ 0 for t ≥ 0. 1 1 1 Proof. Let u˚ ∈ ( ) and, for N ∈ N, deﬁne the operator P : → by + N w w w P f := f e = ( f , f ,..., f , 0,...), f ∈ . N n n 1 2 N n=1 (w) (N ) (w) Then, P u˚ ∈ D(C ) for all N ∈ N, and so, on setting u (t ) = S (t )P u˚,we N + N have t t (N ) (N ) (N ) u (t ) = P u˚ − a u (s) ds + a b u (s) ds, (4.2) N n n j n, j n n 0 0 j =n+1 for n = 1, 2,..., N . Clearly, P u˚ → u˚ in as N →∞, and so, by the continuity (N ) (w) of S (t ),itfollows that u (t ) → u (t ) as N →∞ for all n ∈ N and t ≥ 0. n n (N ) (N ) (w) 2 1 Moreover, if N ≥ N then u (t ) − u (t ) ≥ 0 for all t ≥ 0, since (S (t )) 2 1 t ≥0 (N ) is linear and positive. Similarly, u(t ) − u (t ) ≥ 0 for all N ∈ N and t ≥ 0. Hence, (N ) ∞ (u (t )) is monotone increasing and bounded above by u(t ), and therefore, for N =1 (N ) each ﬁxed n ∈ N, (u (t )) is monotone increasing and bounded above by u (t ). n n N =1 On allowing N →∞ in (4.2), and using the monotone convergence theorem, we obtain t t (N ) u (t ) = u˚ − a u (s) ds + lim a b u (s) ds. n n n n j n, j N →∞ 0 0 j =n+1 From this, we deduce that (N ) lim a b u (s) ds j n, j N →∞ j =n+1 exists, and a further application of the monotone convergence theorem shows that ∞ ∞ (N ) lim a b u (s) ds = a b u (s) ds. j n, j j n, j j N →∞ j =n+1 j =n+1 0 1 Discrete fragmentation systems in weighted spaces Thus, for all u ∈ ( ) , (w) (w) (w) ˚ ˚ ˚ ˚ (S (t )u) = u + −a (S (s)u) + a b (S (s)u) ds. (4.3) n n n n j n, j j j =n+1 (w) It follows that (S (t )u˚) is absolutely continuous with respect to t for each n = 1, 2,... and so (w) (w) (w) S (t )u˚ =−a S (t )u˚ + a b (S (t )u˚) , n ∈ N, (4.4) n j n, j j n n dt j =n+1 for all u˚ ∈ ( ) and almost every t ≥ 0. When u˚ is a general, and therefore not necessarily non-negative, sequence in , we can express u˚ = u˚ − u˚ ∈ . It then follows immediately from the ﬁrst part of + − (w) the proof that u(t ) = S (t )u˚ also satisﬁes (1.1) for almost all t ≥ 0. The last statement of the theorem follows immediately from the positivity of the (w) semigroup (S (t )) . t ≥0 Note that, in general, solutions of (1.1) are not unique; see the discussion in Exam- ple 4.3. We now turn our attention to obtaining a simple representation of the generator (w) (w) (w) (w) (w) G . Although we know that G coincides with A + B on D(A ), and (w) (w) also that u(t ) = S (t )u˚ is the unique classical solution of (3.15)for u˚ ∈ D(G ), (w) we have yet to ascertain an explicit expression that describes the action of G on (w) (w) D(G ). This matter is resolved by the following theorem, which shows that G (w) is a restriction of the maximal operator, G , deﬁned in (1.19). In the speciﬁc case max of X , the result has been obtained from [9, Theorem 6.20], which uses extension [ p] techniques ﬁrst introduced by Arlotti in [4] and which is applied in [8, Theorem 2.1]. We present an alternative proof, which avoids the use of such extensions. (w) Theorem 4.2. Let Assumptions 1.1 and 3.2 hold. Then, for all g ∈ D(G ), we have (w) G g =−a g + a b g , n ∈ N. (4.5) n n j n, j j j =n+1 (w) Proof. It follows from Lemma 2.2 and its proof that, for every g ∈ D(G ), there (w) (w) 1 exist g , g ∈ D(G ) such that g = g − g and f := (I − G )g ∈ ( ) for 1 2 + 1 2 j j + (w) (w) j = 1, 2. This and the linearity of G allow us to assume that g ∈ D(G ) such (w) 1 (w) that f := (I − G )g ∈ ( ) . Deﬁning u(t ) = S (t ) f ,wehavefrom (4.3) that ∞ ∞ (w) −t (w) −t R(1, G ) f = e [S (t ) f ] dt = e u (t ) dt n n 0 0 ∞ t ∞ t −t −t = f − e a u (s) ds dt + e a b u (s) ds dt. n n n j n, j j 0 0 0 0 j =n+1 L. Kerr et al. J. Evol. Equ. By Tonelli’s theorem, we have ∞ t ∞ ∞ −t −t e a u (s) ds dt = e a u (s) dt ds n n n n 0 0 0 s −s (w) = a e u (s) ds = a R(1, G ) f . n n n Using Tonelli’s theorem and the monotone convergence theorem, we obtain ∞ ∞ ∞ t −t (w) e a b u (s) ds dt = a b R(1, G ) f . j n, j j j n, j 0 0 j =n+1 j =n+1 Thus, (w) (w) (w) g = R 1, G f = f − a R 1, G f + a b R 1, G f n n n j n, j n n j j =n+1 (w) (w) (w) = I − G g − a R 1, G f + a b R 1, G f n j n, j n n j j =n+1 (w) = g − G g − a g + a b g , n n n j n, j j j =n+1 and (4.5) follows. We note that the formula (4.5) is independent of the weight w = (w ) . Being n=1 (w) able to express the action of G in this way is important when investigating the full coagulation–fragmentation system, as it enables the fragmentation terms to be (w) described by means of an explicit formula for the operator G . We shall return to this in a subsequent paper. Example 4.3. Let us consider the system u (t ) =−(n − 1)u (t ) + 2 u (t ), t > 0; n j (4.6) j =n+1 u (0) = u˚ , n = 1, 2,..., n n which coincides with (1.1)ifone sets a = n − 1, b = , n, j ∈ N, j > n. (4.7) n n, j j − 1 The system (4.6) models random scission; see, e.g. [25, equation (49)] and [14, equa- tion (10)]. It is easily seen that (1.8) is satisﬁed, and hence, mass is conserved. The example (4.6) is closely related to the example that is studied in [19, §3] and which 1 Discrete fragmentation systems in weighted spaces models random bond annihilation. More precisely, if we denote the operators for the (w) (w) (w) example from [19]by A , B , G etc., then (w) (w) (w) (w) (w) (w) A = A + I, B = B , G = G + I, (w) t (w) and hence S (t ) = e S (t ), t ≥ 0. For the particular case when w = n, n ∈ N, we have similar relations for the operators A , A etc. It follows from [19, Lemma 3.6] 1 1 that every λ> 0 is an eigenvalue of the maximal operator G [i.e. the operator 1,max (w) (λ) (λ) G deﬁned in (1.19)for w = n] with eigenvector g = (g ) where n n n∈N max (λ) g = , n ∈ N. (4.8) (λ + n − 1)(λ + n)(λ + n + 1) The existence of positive eigenvalues of G implies that G is a proper exten- 1,max 1,max sion of G . Note that the domain of G is determined explicitly in [19, Theorem 3.7], 1 1 from which we obtain that D(G ) = f = ( f ) ∈ D(G ) : lim n f = 0 . (4.9) 1 k k∈N 1,max k n→∞ k=n+1 (λ) Using the eigenvectors g from (4.8), we can deﬁne the function (λ) λt (λ) u (t ) := e g , t ≥ 0, which is a solution of the ACP u (t ) = G u(t ), t > 0; u(0) = u˚ (4.10) 1,max (λ) with u˚ = g . On the other hand, since the semigroup (S (t )) is analytic by [19, 1 t ≥0 (λ) Theorem 3.4], the function u(t ) = S (t )g , t ≥ 0, is also a solution of (4.10) and (λ) is distinct from u . This shows that, in general, one does not have uniqueness of solutions of the ACP, (4.10), corresponding to the maximal operator, G , and 1,max hence, also solutions of (1.1) are not unique. (w) More generally, a speciﬁc characterisation of D(G ) is given by Banasiak and Arlotti [9, Theorem 6.20], but this does not lead to an explicit description, such as that obtained in Example 4.3. 5. Analyticity of the fragmentation semigroup In Sect. 3, we established that Assumptions 1.1 and 3.2 are sufﬁcient condi- (w) (w) (w) tions for G = A + B to be the generator of a substochastic C -semigroup, (w) 1 (S (t )) ,on . This enabled us to obtain results on the existence and uniqueness t ≥0 (w) of solutions to (1.17). We now investigate the analyticity of (S (t )) and prove t ≥0 that, given any fragmentation coefﬁcients, it is always possible to construct a weight, (w) (w) w, such that A + B is the generator of an analytic, substochastic C -semigroup 0 L. Kerr et al. J. Evol. Equ. on . This particular result, which is one of the main motivations for carrying out an analysis of the fragmentation system in general weighted spaces, requires a stronger assumption on the weight w. Note that when dealing with analytic semigroups, we use complex versions of the spaces . Assumption 5.1. (i) w ≥ n for all n ∈ N. (ii) There exists κ ∈ (0, 1) such that j −1 w b ≤ κw for all j = 2, 3,.... (5.1) n n, j j n=1 Note that Assumption 5.1 is obtained from Assumption 3.2 by simply replacing κ ∈ (0, 1] with κ ∈ (0, 1). By removing the possibility of κ = 1, we can obtain the following improved version of Theorem 3.4. (w) (w) Theorem 5.2. Let Assumptions 1.1 and 5.1 hold. Then, the operator G = A + (w) (w) 1 B is the generator of an analytic, substochastic C -semigroup, (S (t )) ,on . 0 t ≥0 (w) (w) Proof. Let (T (t )) be as in Lemma 3.1.For α> 0 and f ∈ D(A ) , we obtain t ≥0 + from (3.6) that ! ! (w) (w) ! ! B T (t ) f dt ! ! (w) (w) ! ! ≤ κ A T (t ) f dt α α (w) (w) (w) (w) = κ φ −A T (t ) f dt = κφ − A T (t ) f dt w w 0 0 (w) (w) = κφ − T (t ) f dt = κφ f − T (α) f w w dt (w) = κ f − κT (α) f ≤ κ f . w w w (w) (w) (w) Since κ< 1, it follows from [20, Theorem A.2] that G = A + B is the generator of a positive C -semigroup. The proof of [20, Theorem A.2] establishes (w) that this semigroup is substochastic since κ< 1. Moreover, by Lemma 3.1, A is (w) 1 also the generator of a substochastic C -semigroup, (T (t )) ,on , and a routine 0 t ≥0 calculation shows that ! ! 1 1 (w) ! ! R(λ, A ) f = w | f |≤ f ,λ ∈ C\R with Re λ> 0, n n w |λ + a | | Im λ| n=1 1 (w) for all f ∈ . Therefore, by [17, Theorem II.4.6], (T (t )) is an analytic semi- t ≥0 (w) (w) (w) group. Also, the positivity of (S (t )) implies that A +B is resolvent positive. t ≥0 (w) Hence, by [2, Theorem 1.1], (S (t )) is analytic. t ≥0 1 Discrete fragmentation systems in weighted spaces Remark 5.3. (i) Although Assumption 5.1 is never satisﬁed when (1.8) holds and w = n for all n ∈ N, this does not rule out the possibility of an analytic fragmentation semigroup on X existing. Indeed, the semigroup (S (t )) in [1] 1 t ≥0 Example 4.3 is analytic, which follows from [19, Theorem 3.4] as mentioned above. (ii) If there exists λ > 0 such that (1.5) holds with λ ≥ λ for all j ≥ 2(which 0 j 0 corresponds to a ‘uniform’ mass loss case), then Assumption 5.1 immediately holds with w = n for all n ∈ N, and κ = 1 − λ . n 0 The following lemma gives sufﬁcient conditions under which Assumption 5.1 holds. Lemma 5.4. Let w be such that w w n n+1 1 ≤ ≤ δ for all n ∈ N, (5.2) n n + 1 where δ ∈ (0, 1). Moreover, let (1.4) hold. Then, Assumption 5.1 is satisﬁed with κ = δ. Proof. Since w w w n j j j −n ≤ δ ≤ δ for all n = 1,..., j − 1, n j j it follows that j −1 j −1 j −1 w w n j w b = nb ≤ δ nb ≤ δw n n, j n, j n, j j n j n=1 n=1 n=1 for j = 2, 3,..., where (1.4) is used to obtain the last inequality. Since δ ∈ (0, 1),the result follows immediately. This leads to the main result of this section. Theorem 5.5. For any given fragmentation coefﬁcients for which Assumption 1.1 ∞ (w) (w) holds, we can always ﬁnd a weight, w = (w ) , such that A + B is the n=1 generator of an analytic, substochastic C -semigroup on . If, in addition, (1.4) holds, we can choose w = r with arbitrary r > 2 and κ = 2/r so that (5.1) holds. Proof. For the ﬁrst statement, note that we can choose w ≥ n iteratively so that (5.1) is satisﬁed. The claim then follows from Theorem 5.2. Now, assume that (1.4) holds. Let r > 2, w = r for n ∈ N, and δ = 2/r, which satisﬁes δ< 1. Then, w ≥ n and n+1 n n n w 2 r 2r 2r r w n+1 n δ = · = ≥ = = , n + 1 r n + 1 n + 1 n + n n n which shows that (5.2) is satisﬁed. Hence, Lemma 5.4 implies that Assumption 5.1 is fulﬁlled. L. Kerr et al. J. Evol. Equ. As mentioned earlier, analytic semigroups have a number of desirable properties, and Theorem 5.5 will play an important role when we investigate the full coagulation– fragmentation system in a subsequent paper. In particular, Theorem 5.5 will enable us to relax the usual assumptions that are imposed on the coagulation rates in order to obtain the existence and uniqueness of solutions to the full coagulation–fragmentation system. It should be noted that a condition that is equivalent to Assumption 5.1 has previously been used as a condition for analyticity in the mass-conserving case by Banasiak; see [8, Theorem 2.1]. However, the choice of weights in [8] is restricted to w = n , p > 1, and Assumption 5.1 need not be satisﬁed for these weights for any p > 1as the following example shows. Example 5.6. Consider the mass-conserving case where a cluster of mass n breaks into two clusters, with respective masses 1 and n −1. The corresponding fragmentation coefﬁcients take the form b = 2; b = b = 1, j ≥ 3; b = 0, 2 ≤ n ≤ j − 2. (5.3) 1,2 1, j j −1, j n, j For the choice a = 0; a = n, n ≥ 2; w = n , n ∈ N; p ≥ 1, 0 n n it is proved in [7, Theorem 3] (for p = 1) and [8, Theorem A.3] (for p > 1) that (w) the semigroup generated by G is not analytic. On the other hand, Theorem 5.5 (w) guarantees the existence of exponentially growing weights w such that G = (w) (w) A + B generates an analytic semigroup. It is easy to show that for this particular example one can also choose powers of 2, namely w = 1 and w = 2 for n ≥ 2, in 1 n which case κ = 5/8. 6. Asymptotic behaviour of solutions There have been several earlier investigations into the long-term behaviour of solu- tions to the mass-conserving fragmentation system (1.1), when (1.8) holds. In partic- ular, the case of mass-conserving binary fragmentation is dealt with in [15], where it is shown that, under suitable assumptions, the unique solution emanating from u˚ must converge in the space X to the expected steady-state solution M (u˚)e , where [1] 1 1 M (u˚) and e are given by (1.6) and (3.18), respectively. This was followed by [7], and 1 1 [12] where, once again, the expected long-term steady-state behaviour is established, but now for the mass-conserving multiple fragmentation system. More speciﬁcally, in [7], a semigroup-based approach is used to prove that, for any u˚ ∈ X , [1] lim S (t )u˚ − M (u˚)e = 0 if and only if a > 0 for all n = 2, 3,.... 1 1 [1] n t →∞ 1 Discrete fragmentation systems in weighted spaces That the corresponding result is also valid in the higher moment spaces X , p > 1, [ p] is established in [12], and, under additional assumptions on the fragmentation coefﬁ- cients, it is shown in [12, Theorem 4.3] that there exist constants L > 0 and α> 0 such that the fragmentation semigroup (S (t )) on X , p > 1, satisﬁes p t ≥0 [ p] −αt S (t )u˚ − M (u˚)e ≤ Le u˚ , (6.1) p 1 1 [ p] [ p] for all u˚ ∈ X . It follows from [3] that the fragmentation semigroup (S (t )) has the [ p] p t ≥0 asynchronous exponential growth (AEG) property (with λ = 0in [3, equation (3)], i.e. with trivial growth). The assumptions required in [12] to prove that (6.1) holds in some X space are somewhat technical and not straightforward to check. Moreover, [ p] no information on the size of the constant α, and hence the exponential rate of decay to the steady state is provided. Our aim in this section is to address these issues. Working within the framework of more general weighted spaces, we study the long-term dynamics of solutions in both the mass-conserving and mass loss cases. When mass is conserved, we establish simpler conditions under which the fragmentation semigroup (w) 1 (S (t )) satisﬁes an inequality of the form (6.1) on some space , and also t ≥0 quantify α. We begin by considering the general fragmentation system (1.1), where the coefﬁ- (w) (w) (w) cients a and b satisfy Assumption 1.1, and recall that G = A + B is the n n, j (w) 1 generator of a substochastic C -semigroup, (S (t )) ,on whenever Assump- 0 t ≥0 (w) (w) (w) tion 3.2 holds. Furthermore, (S (t )) is analytic, with generator A + B when t ≥0 the more restrictive Assumption 5.1 is satisﬁed. Theorem 6.1. Let Assumptions 1.1 and 3.2 hold. (i) Then, (w) lim S (t )u˚ =0(6.2) t →∞ for all u˚ ∈ if and only if a > 0 for all n ∈ N. (ii) If, additionally, we choose w such that Assumption 5.1 is satisﬁed, and set a := inf a , then n∈N (w) −(1−κ)a t S (t )≤ e , (6.3) and hence, if a > 0 and α ∈[0,(1 − κ)a ), we have 0 0 αt (w) 1 lim e S (t )u˚ = 0 for every u˚ ∈ . (6.4) t →∞ If α> a , then (6.4) does not hold. In particular, if a = 0, then (6.4) does not 0 0 hold for any α> 0. Proof. (i) First assume that a > 0 for all n ∈ N.Let u ∈ , and, as in Sect. 4,let P u˚ = (u˚ , u˚ ,..., u˚ , 0,...), N ∈ N. For each ﬁxed n ∈ N, we know from (3.22) N 1 2 N (w) that (S (t )e ) = s (t ) = 0for m > n. Furthermore, (s , s ,..., s ), with n m m,n 1,n 2,n n,n the identiﬁcation (3.21), is the unique solution of the n-dimensional system (3.19). Our assumption on the coefﬁcients a means that all eigenvalues of the matrix associated n L. Kerr et al. J. Evol. Equ. with (3.19) are negative. It follows that s (t ) → 0as t →∞ for m = 1,..., n, and m,n therefore (w) lim S (t )e = lim w s (t ) = 0, n w m m,n t →∞ t →∞ m=1 for all n ∈ N. This in turn implies that (w) (w) S (t )P u˚ ≤ |u˚ |S (t )e →0as t →∞, N w n n w n=1 for each N ∈ N. Given any ε> 0, we can always ﬁnd N ∈ N and t > 0 such that ε ε (w) u˚ − P u˚ < and S (t )P u˚ < for all t ≥ t . N w N w 0 2 2 Then, ! ! (w) (w) (w) ! ! S (t )u˚ ≤ S (t ) u˚ − P u˚ +S (t )P u˚ w N N w (w) ≤u˚ − P u˚ +S (t )P u˚ <ε for all t ≥ t , N w N w 0 which establishes (6.2). On the other hand, suppose that a = 0for some N ∈ N. Then, we have that the (w) ∞ unique solution of (1.17), with u˚ = e ,is u(t ) = S (t )e = (s (t )) . Since N N m,N m=1 −a t s (t ) = e = 1, it is clear that u(t ) → 0as t →∞. N ,N 1 (w) (w) (ii) Now, let Assumption 5.1 hold and let u˚ ∈ ( ) . From Theorem 5.2, A +B (w) 1 generates an analytic, substochastic C -semigroup, (S (t )) ,on , and u(t ) = 0 t ≥0 (w) S (t )u˚ is the unique, non-negative classical solution of (1.17). Let t > 0. Using (3.5), we obtain that (w) (w) φ u(t ) = φ u (t ) = φ A u(t ) + φ B u(t ) w w w w dt (w) (w) ≤ φ A u(t ) − κφ A u(t ) w w =−(1 − κ) w a u (t ) n n n n=1 ≤−(1 − κ)a φ u(t ) . 0 w Therefore, −(1−κ)a t (w) −(1−κ)a t 0 0 φ u(t ) ≤ φ (u˚)e and hence S (t )u˚ ≤ e u˚ , w w w w (w) and (6.3) then follows from the positivity of (S (t )) and [9, Proposition 2.67]. If t ≥0 a > 0 and α ∈[0,(1 − κ)a ), then (6.4) holds. 0 0 On the other hand, if we choose α> a , then there exists N ∈ N such that a <α, 0 N (w) −a t −αt in which case (S (t )e ) = e > e for t > 0, and so N N αt (w) αt (w) αt −αt e S (t )e ≥ e w S (t )e > e w e = w . N w N N N N Hence, (6.4) cannot hold for any α> a . 0 1 Discrete fragmentation systems in weighted spaces Remark 6.2. When the assumptions of Theorem 6.1 are satisﬁed and a > 0 for all n ∈ N, then (6.2) shows that the only equilibrium solution of (1.17)is u(t ) ≡ 0, and this equilibrium is a global attractor for the system. On the other hand, if a = 0for at least one n ∈ N, then u(t ) ≡ 0 is not a global attractor. We now examine the mass-conserving case and assume that (1.8) holds. Note that, in this mass-conserving case, the fragmentation semigroup (S (t )) is stochastic 1 t ≥0 on the space X . Our aim is to establish an version of the results obtained in [1] (w) [7,12,15]. To this end, we recall the matrix representation of S (t ) given by (3.22) (w) and also deﬁne a sequence space Y , and its norm · (w),by (w) ∞ ∞ 1 Y = f = ( f ) : f = ( f ) ∈ and f = w | f |, (w) n n n n n=2 n=1 w Y n=2 (w) 1 1 respectively. Clearly, Y is a weighted space and can be identiﬁed with , where (w) 1 w = w for n ∈ N. Moreover, we deﬁne the embedding operator J : Y → n n+1 by Jf = (0, f , f ,...) for all f ∈ . 2 3 1 ∞ Lemma 6.3. Let α ≥ 0 and f ∈ be ﬁxed, and deﬁne f := ( f ) .IfAssump- n=2 tions 1.1, 3.2 and (1.8) hold, then (w) (w) (w) ˜ ˜ S (t ) f ≤S (t ) f − M ( f )e ≤ (w + 1)S (t ) f (6.5) (w) (w) 1 1 w 1 Y Y (22) (22) for all t ≥ 0. Proof. It follows from (3.22) that (w) (w) (w) ˜ ˜ S (t ) f = f + S (t ) f e + JS (t ) f . (6.6) 1 1 (12) (22) From this, we deduce that (w) (w) (w) ˜ ˜ S (t ) f − M ( f )e = w f + S (t ) f − M ( f ) +S (t ) f (6.7) (w) 1 1 w 1 1 1 (12) (22) and so (w) (w) S (t ) f − M ( f )e ≥S (t ) f (w) , 1 1 w (22) which is the ﬁrst inequality in (6.5). On the other hand, from Proposition 2.3 (i) and the stochasticity of (S (t )) on 1 t ≥0 (w) X , we know that M (S (t ) f ) = M (S (t ) f ) = M ( f ).Using (6.6), we obtain [1] 1 1 1 1 L. Kerr et al. J. Evol. Equ. (w) (w) ˜ ˜ f + S (t ) f − M ( f ) = M ( f ) − M f + S (t ) f e 1 1 1 1 1 1 (12) (12) (w) (w) = M S (t ) f − M f + S (t ) f e 1 1 1 1 (12) (w) (w) ≤ M S (t ) f − f + S (t ) f e 1 1 1 (12) (w) (w) ≤ φ S (t ) f − f + S (t ) f e w 1 1 (12) (w) = φ (JS (t ) f (22) ! ! (w) ! ˜! = S (t ) f . (w) (22) Y The second inequality in (6.5) then follows from (6.7). We are now in a position to prove the main theorem of this section. The ﬁrst part (w) 1 1 conﬁrms that S (t )u˚ → M (u˚)e in as t →∞, for all u˚ ∈ , provided that 1 1 w w Assumption 3.2 holds and the fragmentation rates, a , are positive for all n ≥ 2. In the second part, which deals with quantifying the rate of convergence to equilibrium, the fragmentation coefﬁcients are assumed additionally to be bounded below by a positive constant, and Assumption 3.2 is strengthened to Assumption 5.1. In this case, (w) the decay to zero of S (t )u˚ − M (u˚)e is shown to occur at an exponential 1 1 w rate, deﬁned explicitly in terms of the rate coefﬁcients and the constant κ ∈ (0, 1) in Assumption 5.1. Theorem 6.4. Let Assumptions 1.1 and 3.2, and (1.8) hold and let M be as in (1.6). (i) We have (w) lim S (t )u˚ − M (u˚)e = 0 (6.8) 1 1 w t →∞ for all u˚ ∈ if and only if a > 0 for all n ≥ 2. (ii) Choose w such that Assumption 5.1 holds and let a := inf a . Then, for 0 n∈N:n≥2 n all u˚ ∈ , (w) −(1−κ) a t S (t )u˚ − M (u˚)e ≤ (w + 1)e u˚ , (6.9) 1 1 w 1 w and so αt (w) lim e S (t )u˚ − M (u˚)e = 0, (6.10) 1 1 w t →∞ whenever a > 0 and α ∈[0,(1 − κ) a ). 0 0 Equation (6.10) does not hold for any α> a . In particular, if a = 0, then 0 0 (6.10) does not hold for any α> 0. Proof. Removing the equation for u from (1.1) leads to a reduced fragmentation (w) 1 system that can be formulated as an ACP in Y = , where, as before, w = w n n+1 for all n ∈ N. The fragmentation coefﬁcients, ( a ) and (b ) , associated n n, j n, j ∈N:n< j n=1 with the reduced system are given by a = a and b = b . Clearly, a ≥ 0 n n+1 n, j n+1, j +1 n 1 Discrete fragmentation systems in weighted spaces and b ≥ 0 for all n, j ∈ N and b = 0if n ≥ j, and w = w ≥ n + 1 > n for n, j n, j n n+1 all n ∈ N. Moreover, for j = 2, 3,..., j −1 j −1 j j w b = w b = w b ≤ w b n n, j n+1 n+1, j +1 k k, j +1 k k, j +1 n=1 n=1 k=2 k=1 ≤ κw = κw . j +1 j Hence, Assumptions 1.1 and 3.2 are satisﬁed by w , a and b , and it follows from n n, j Theorem 3.4 and (3.22) that associated with the reduced system is a substochastic (w) C -semigroup on Y , which can be represented by the inﬁnite matrix ⎡ ⎤ ⎡ ⎤ − a t −a t 1 2 e s (t ) s (t ) ··· e s (t ) s (t ) ··· 1,2 1,3 1,2 1,3 − a t −a t ⎢ 2 ⎥ ⎢ 3 ⎥ 0 e s (t ) ··· 0 e s (t ) ··· 2,3 2,3 ⎢ ⎥ ⎢ ⎥ ⎢ − a t ⎥ = ⎢ −a t ⎥ , (6.11) 3 4 00 e ··· 00 e ··· ⎣ ⎦ ⎣ ⎦ . . . . . . . . . . . . . . . . . . . . . . . . where, for all n ∈ N, m = 1,..., n − 1, t ≥ 0, s (t ) is the unique solution of m,n s (t ) =− a s (t ) + a b s (t ) m m,n j m, j j,n m,n j =m+1 n+1 =−a s (t ) + a b s (t ). m+1 m,n k m+1,k k−1,n k=m+2 An inspection of (3.19), together with (3.21), shows that s (t ) = s (t ) for all m,n m+1,n+1 (w) n ∈ N, m = 1,..., n − 1, t ≥ 0, and therefore the substochastic semigroup on Y is (w) (w) given by (S (t )) , where (S (t )) is the inﬁnite matrix that features in (3.22). t ≥0 t ≥0 (22) (22) (i) Let u˚ = (u˚ , u˚ ,...) for each u˚ ∈ . From Theorem 6.1, we deduce that 2 3 ! ! ! ! (w) (w) ! ! ! ! lim S (t )u˚ = lim S (t )u˚ = 0, (w) (22) (22) Y w t →∞ t →∞ if and only if a > 0 for all n ≥ 2, and the result is then an immediate consequence of Lemma 6.3. (ii) The calculations above show that, when Assumption 3.2 holds for w and the coefﬁcients (b ), it is also satisﬁed by w and (b ) with exactly the same value of n, j n, j κ. Therefore, from Theorem 6.1, (w) −(1−κ) a t S (t )≤ e , (22) and (6.9) follows immediately from Lemma 6.3. Moreover, if a > 0 and α ∈ [0,(1 − κ) a ), then we obtain (6.10). If α> a , then, from Theorem 6.1,the result ! ! ! ! (w) (w) αt αt ! ! ! ! lim e S (t )u˚ = lim e S (t )u˚ = 0, (w) (22) (22) Y w t →∞ t →∞ does not hold for all u˚ ∈ . Hence, from Lemma 6.3,(6.10) does not hold if α> a . 0 L. Kerr et al. J. Evol. Equ. Remark 6.5. When the assumptions of Theorem 6.4 are satisﬁed, then it follows from (3.22) that u = Me is an equilibrium solution of the mass-conserving fragmentation M 1 system for all M ∈ R. In addition, the basin of attraction for u is given by {u˚ ∈ : M (u˚) = M } provided that the assumptions of Theorem 6.4 hold and a > 0 for all 1 n n ≥ 2. On the other hand, if a = 0for some N ≥ 2, then Me is also an equilibrium N N solution for every M ∈ R. 7. Sobolev towers In this section, we use a Sobolev tower construction to obtain existence and unique- ness results relating to the pure fragmentation system for a larger class of initial con- ditions. Sobolev towers appear to have been ﬁrst applied to the discrete fragmentation system (1.1)in[19], where the authors examine a speciﬁc example and use Sobolev towers to explain an apparent non-uniqueness of solutions. As we demonstrate below, the theory of Sobolev towers is applicable to more general fragmentation systems and, in the following, the only restrictions that are imposed are that the fragmentation coef- ﬁcients satisfy Assumption 1.1, and also that a weight, w = (w ) , has been chosen n=1 (w) (w) (w) so that Assumption 5.1 holds. These restrictions imply that G = A + B is the (w) 1 generator of an analytic, substochastic C -semigroup, (S (t )) ,on .Let ω be 0 t ≥0 0 (w) (w) the growth bound of (S (t )) . Choosing μ>ω , we rescale (S (t )) to obtain t ≥0 0 t ≥0 (w) −μt (w) an analytic semigroup, (S (t )) = (e S (t )) , with a strictly negative t ≥0 t ≥0 (w) (w) (w) (w) 1 growth bound. The generator of (S (t )) is G = G −μI.Weset X = , t ≥0 (w) (w) (w) (w) (w) (w) · := · , S (t ) = S (t ), S (t ) = S (t ), and G = G . 0 w 0 0 0 (w) As described in [17, §II.5(a)], (S (t )) can be used to construct a Sobolev t ≥0 (w) tower, (X ) ,via n n∈N ! ! (w) (w) n (w) n (w) n ! ! X := D (G ) , · ; f = (G ) f , f ∈ D (G ) , n ∈ N. n n (w) For each n ∈ N, X is referred to as the Sobolev space of order n associated with the (w) (w) (w) (w) (w) semigroup (S (t )) . We also deﬁne the operator G : X ⊇ D(G ) → X t ≥0 n n n n (w) to be the restriction of G to (w) (w) (w) (w) (w) n+1 (w) D(G ) = f ∈ X : G f ∈ X = D (G ) = X , n n n n+1 for each n ∈ N. Sobolev spaces of negative order, −n, n ∈ N, are deﬁned recursively by ! ! (w) (w) (w) (w) −1 ! ! X = X , · ; f = (G ) f , f ∈ X , −n −n −n −n+1 −n+1 −n+1 −n+1 (7.1) where (X, ·) denotes the completion of the normed vector space (X, ·). Oper- (w) ators G can then be obtained in a similar recursive manner for each n ∈ N, −n (w) (w) (w) (w) with G deﬁned as the unique extension of G from D(G ) = X to −n −n+1 −n+1 −n+2 (w) (w) D(G ) = X ;see [17, §II.5(a)]. −n −n+1 1 Discrete fragmentation systems in weighted spaces (w) From [17, §II.5(a)], it follows that G is the generator of an analytic, substochastic (w) (w) (w) C -semigroup, (S (t )) ,on X for all n ∈ Z, where S (t ) is the unique, 0 n t ≥0 n −n (w) (w) (w) continuous extension of S (t ) from X to X for each t ≥ 0 and n ∈ N. Since −n (w) −μt (w) S (t ) = e S (t ), we also obtain the analytic, substochastic C -semigroup, (w) (w) (w) (w) μt (S (t )) , deﬁned on X by S (t ) = e S (t ). More generally, it is known t ≥0 −n −n −n −n (w) (w) (w) (w) that S (t ) is the unique, continuous extension of S (t ) from X to X when n m m n (w) (w) m, n ∈ Z with m ≥ n. The analyticity of (S (t )) on X , also enables us to n t ≥0 n prove the following key result. (w) (w) (w) Lemma 7.1. Let u˚ ∈ X for some ﬁxed n ∈ Z. Then, S (t )u˚ ∈ X for all n n m m ≥ n and t > 0. (w) (w) (w) Proof. It is obvious that S (t )u˚ ∈ X for all t ≥ 0 and u˚ ∈ X . Also, if n n n (w) (w) S (t )u˚ ∈ X for some m ≥ n and all t > 0, then, on choosing t ∈ (0, t ),wehave n m 0 (w) (w) (w) (w) (w) S (t )u˚ = S (t − t )S (t )u˚ ∈ D(G ) = X , 0 0 n m n m m+1 (w) (w) (w) where we have used the fact that S (t ) and S (t ) coincide on X together with n m m (w) the analyticity of S (t ). The result then follows by induction. We can now prove the following result regarding the solvability of (1.17). Theorem 7.2. Let Assumptions 1.1 and 5.1 hold. Further, let n ∈ N. Then, the ACP (w) 1 1 (1.17) has a unique, non-negative solution u ∈ C ((0, ∞), ) ∩C ([0, ∞), X ) for w −n (w) (w) all u˚ ∈ (X ) . This solution is given by u(t ) = S (t )u˚, t ≥ 0. −n −n (w) (w) μt Proof. Let u˚ ∈ (X ) and let u(t ) = S (t )u˚ = e v(t ), t ≥ 0, where v(t ) = −n −n (w) (w) (w) S (t )u˚. Then, v ∈ C ((0, ∞), X ) ∩ C ([0, ∞), X ) is the unique classical −n −n −n solution of (w) v (t ) = G v(t ), t > 0; v(0) = u˚. (7.2) −n (w) (w) (w) (w) Also, from Lemma 7.1, S (t )u˚ ∈ X = D(G ) for all t > 0. Since (S (t )) t ≥0 −n −n (w) (w) coincides with (S (t )) on D(G ),itfollows that t ≥0 (w) (w) (w) S (t )u˚ = S (t − t )S (t )u˚, where t ∈ (0, t ). 0 0 0 −n −n Consequently, (w) (w) (w) (w) (w) (w) S (t )u˚ = G S (t − t )S (t )u˚ = G S (t )u˚, t > 0, 0 0 −n −n −n dt (w) where the derivative is with respect to the norm on X = . This establishes that (w) 1 1 u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ) and also that u satisﬁes (1.17). The non- w −n negativity of u follows from the substochasticity of the semigroups. For uniqueness, we observe ﬁrst that the construction of the Sobolev tower ensures (w) (w) (w) (w) that X is continuously embedded in X . Moreover, G is the restriction of G −n −n 0 L. Kerr et al. J. Evol. Equ. (w) (w) (w) 1 1 to X = D(G ). Consequently, if u , u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ) 1 2 w −n −μt both satisfy (1.17), and we set v (t ) = e u (t ), i = 1, 2, then the difference v − v i i 1 2 is the unique classical solution of (7.2) with u˚ = 0, and so v = v , from which it 1 2 follows that u = u . 1 2 Finally, we make the following remark on the solvability of (3.2). Remark 7.3. For ﬁxed n ∈ N, the previous theorem establishes that the ACP (1.17) (w) 1 1 has a unique, non-negative solution u ∈ C ((0, ∞), ) ∩ C ([0, ∞), X ),given w −n (w) (w) ˚ ˚ by u(t ) = S (t )u, for all u ∈ (X ) , provided that Assumptions 1.1 and 5.1 −n −n are satisﬁed. Recalling that we also assume that w ≥ n for all n ∈ N,wehave that is continuously embedded in X , and from this we deduce that if u(t ) is [1] differentiable with respect to the norm on then it is also differentiable with respect to the norm on X , and the derivatives coincide. Since A + B is an extension of [1] 1 1 (w) (w) (w) (w) G = A + B , we conclude that u(t ) = S (t )u˚ also satisﬁes (3.2). −n Acknowledgements L. Kerr gratefully acknowledges the support of The Carnegie Trust for the Univer- sities of Scotland. All authors would like to thank the referees for their very helpful comments. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/ by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations. REFERENCES [1] M. Aizenman and T. A. Bak, Convergence to equilibrium in a system of reacting polymers. Comm. Math. Phys. 65 (1979), 203–230 [2] W. Arendt and A. Rhandi, Perturbation of positive semigroups. Arch. Math. (Basel) 56 (1991), 107–119 [3] O. Arino, Some spectral properties for the asymptotic behavior of semigroups connected to popu- lation dynamics. SIAM Rev. 34 (1992), 445–476 [4] L. 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Nhumaio, S. T. Sonne, M. Gunsing, J. Straatsma, M. Verschueren, M. Sibeijn, G. Schulte, U. Fritsching, K. Bauckhage, C. Tropea, M. Sommerfeld, A. P. Watkins, A. J. Yule and H. Schønfeldt, Simulation of agglomeration in spray drying installations: the EDECAD project. Drying Technology 22 (2004), 1403–1461 [22] J. Voigt, On substochastic C semigroups and their generators. In: Proceedings of the Conference on Mathematical Methods Applied to Kinetic Equations (Paris, 1985), Transport Theory Statist. Phys., no. 16, pp. 453–466 (1987) [23] J. Wells, Modelling coagulation in industrial spray drying: an efﬁcient one-dimensional population balance approach. Ph.D. thesis, University of Strathclyde, Department of Mathematics and Statistics (2018) [24] R. M. Ziff, Kinetics of polymerization. J. Statist. Phys. 23 (1980), 241–263 [25] R. M. Ziff and E. D. McGrady, The kinetics of cluster fragmentation and depolymerisation. J. Phys. A 18 (1985), 3027–3037 Lyndsay Kerr, Wilson Lamb and Matthias Langer Department of Mathematics and Statistics University of Strathclyde 26 Richmond Street Glasgow G1 1XH UK E-mail: m.langer@strath.ac.uk Lyndsay Kerr E-mail: lyndsay.kerr@strath.ac.uk Wilson Lamb E-mail: w.lamb@strath.ac.uk

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