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Dirk Horstmann, M. Winkler (2005)
Boundedness vs. blow-up in a chemotaxis systemJournal of Differential Equations, 215
J. Lankeit (2014)
Eventual smoothness and asymptotics in a three-dimensional chemotaxis system with logistic sourceJournal of Differential Equations, 258
Youshan Tao (2011)
Boundedness in a chemotaxis model with oxygen consumption by bacteriaJournal of Mathematical Analysis and Applications, 381
E. Keller, L. Segel (1970)
Initiation of slime mold aggregation viewed as an instability.Journal of theoretical biology, 26 3
H. Amann (1993)
Function Spaces, Differential Operators and Nonlinear Analysis
Elio Espejo, A. Stevens, J. Velázquez (2009)
Simultaneous finite time blow-up in a two-species model for chemotaxis, 29
Stanca Ciupe, Virginia Tech, Patrick Nelson (2021)
Mathematical biologyActa Applicandae Mathematica, 23
J. Tello, M. Winkler (2007)
A Chemotaxis System with Logistic SourceCommunications in Partial Differential Equations, 32
J. Zheng (2015)
Boundedness of solutions to a quasilinear parabolic-elliptic Keller-Segel system with logistic sourceJ. Differ. Equ., 259
M. Winkler (2010)
Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivityMathematische Nachrichten, 283
R. Kowalczyk, Zuzanna Szymanska (2008)
On the global existence of solutions to an aggregation modelJournal of Mathematical Analysis and Applications, 343
Ping Liu, Junping Shi, Zhian Wang (2013)
Pattern Formation of the Attraction-Repulsion Keller-Segel SystemDiscrete and Continuous Dynamical Systems-series B, 18
Dirk Horstmann (2011)
Generalizing the Keller–Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting SpeciesJournal of Nonlinear Science, 21
Jia Liu, Zhian Wang (2012)
Classical solutions and steady states of an attraction–repulsion chemotaxis in one dimensionJournal of Biological Dynamics, 6
M. Winkler (2014)
How Far Can Chemotactic Cross-diffusion Enforce Exceeding Carrying Capacities?Journal of Nonlinear Science, 24
C. Conca, Elio Espejo, K. Vilches (2011)
Remarks on the blowup and global existence for a two species chemotactic Keller–Segel system in 2European Journal of Applied Mathematics, 22
Xinru Cao, Sining Zheng (2014)
Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic sourceMathematical Methods in the Applied Sciences, 37
Youshan Tao (2014)
Boundedness in a two-dimensional chemotaxis-haptotaxis systemarXiv: Analysis of PDEs
Daniel Henry (1989)
Geometric Theory of Semilinear Parabolic Equations
Yan Li, Yuxiang Li (2014)
Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two speciesNonlinear Analysis-theory Methods & Applications, 109
Youshan Tao, Zhian Wang (2013)
Competing effects of attraction vs. repulsion in chemotaxisMathematical Models and Methods in Applied Sciences, 23
M. Delgado, I. Gayte, C. Morales-Rodrigo, A. Suárez (2010)
An angiogenesis model with nonlinear chemotactic response and flux at the tumor boundaryNonlinear Analysis-theory Methods & Applications, 72
Tomasz Cieślak, M. Winkler (2008)
Finite-time blow-up in a quasilinear system of chemotaxisNonlinearity, 21
Elio Espejo, Takashi Suzuki (2014)
Global existence and blow-up for a system describing the aggregation of microgliaAppl. Math. Lett., 35
José Castillo, M. Winkler (2012)
Stabilization in a two-species chemotaxis system with a logistic sourceNonlinearity, 25
J. Lankeit (2014)
Chemotaxis can prevent thresholds on population densityarXiv: Analysis of PDEs
M. Winkler, Kianhwa Djie (2010)
Boundedness and finite-time collapse in a chemotaxis system with volume-filling effectNonlinear Analysis-theory Methods & Applications, 72
M. Winkler (2014)
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampeningJournal of Differential Equations, 257
M. Winkler (2010)
Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic SourceCommunications in Partial Differential Equations, 35
Liangchen Wang, Chunlai Mu, Pan Zheng (2014)
On a quasilinear parabolic–elliptic chemotaxis system with logistic sourceJournal of Differential Equations, 256
T. Nagai (2001)
Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domainsJournal of Inequalities and Applications, 2001
Jia Hu, Qi Wang, Jingyue Yang, Lulu Zhang (2014)
Global existence and steady states of a two competing species Keller-Segel chemotaxis modelarXiv: Analysis of PDEs
Chunlai Mu, Liangchen Wang, Pan Zheng, Qingna Zhang (2013)
Global existence and boundedness of classical solutions to a parabolic–parabolic chemotaxis systemNonlinear Analysis-real World Applications, 14
Ke Lin, Chunlai Mu, Liangchen Wang (2015)
Large-time behavior of an attraction–repulsion chemotaxis systemJournal of Mathematical Analysis and Applications, 426
P. Bassanini, A. Elcrat (1997)
Elliptic Partial Differential Equations of Second Order
Qingshan Zhang, Yuxiang Li (2015)
Global boundedness of solutions to a two-species chemotaxis systemZeitschrift für angewandte Mathematik und Physik, 66
M. Hieber, J. Prüss (1997)
Heat kernels and maximal L p − L q $L^{p}-L^{q}$ estimate for parabolic evolution equationsCommun. Partial Differ. Equ., 22
Pan Zheng, Chunlai Mu, Xuegang Hu (2014)
Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic sourceDiscrete and Continuous Dynamical Systems, 35
Xinru Cao (2014)
Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic sourceJournal of Mathematical Analysis and Applications, 412
Youshan Tao, M. Winkler (2014)
Energy-type estimates and global solvability in a two-dimensional chemotaxis–haptotaxis model with remodeling of non-diffusible attractantJournal of Differential Equations, 257
P. Biler, W. Hebisch, T. Nadzieja (1994)
The Debye system: existence and large time behavior of solutionsNonlinear Analysis-theory Methods & Applications, 23
H. Amann (1993)
Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems
Koichi Osaki, T. Tsujikawa, A. Yagi, M. Mimura (2002)
Exponential attractor for a chemotaxis-growth system of equationsNonlinear Analysis-theory Methods & Applications, 51
Xinru Cao (2015)
Boundedness in a three-dimensional chemotaxis–haptotaxis modelZeitschrift für angewandte Mathematik und Physik, 67
Christian Stinner, J. Tello, Michael Winkler (2013)
Competitive exclusion in a two-species chemotaxis modelJournal of Mathematical Biology, 68
M. Winkler (2010)
Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel modelJournal of Differential Equations, 248
C. Yang, Xinru Cao, Zhaoxin Jiang, Sining Zheng (2015)
Boundedness in a quasilinear fully parabolic Keller-Segel system of higher dimension with logistic sourcearXiv: Analysis of PDEs
Pan Zheng, Chunlai Mu, Xuegang Hu, Ya Tian (2015)
Boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic sourceJournal of Mathematical Analysis and Applications, 424
Hai-yang Jin (2015)
Boundedness of the attraction–repulsion Keller–Segel systemJournal of Mathematical Analysis and Applications, 422
Khadijeh Baghaei, M. Hesaaraki (2013)
Global existence and boundedness of classical solutions for a chemotaxis model with logistic sourceComptes Rendus Mathematique, 351
N. Alikakos (1979)
LP Bounds of solutions of reaction-diffusion equationsCommunications in Partial Differential Equations, 4
M. Winkler (2011)
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel systemarXiv: Analysis of PDEs
Youshan Tao, M. Winkler (2015)
Boundedness vs.blow-up in a two-species chemotaxis system with two chemicalsDiscrete and Continuous Dynamical Systems-series B, 20
Yung-Sze Choi, Zhian Wang (2010)
Prevention of blow-up by fast diffusion in chemotaxisJournal of Mathematical Analysis and Applications, 362
Hieber Matthias, Pruss Jan (1997)
Heat kernels and maximal lp—lqestimates for parabolic evolution equationsCommunications in Partial Differential Equations, 22
N.D. Alikakos (1979)
L p $L^{p}$ bounds of solutions of reaction-diffusion equationsCommun. Partial Differ. Equ., 4
Ke Lin, Chunlai Mu, Liangchen Wang (2015)
Boundedness in a two‐species chemotaxis systemMathematical Methods in the Applied Sciences, 38
P. Biler, Elio Espejo, Ignacio Guerra (2012)
Blowup in higher dimensional two species chemotactic systemsCommunications on Pure and Applied Analysis, 12
Pan Zheng, Chunlai Mu (2015)
Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic sourceMathematische Nachrichten, 288
M. Winkler (2008)
Chemotaxis with logistic source : Very weak global solutions and their boundedness propertiesJournal of Mathematical Analysis and Applications, 348
K. Baghaei, M. Hesaaraki (2013)
Global existence and boundedness of classical solutions for a chemotaxis model with logistic sourceC. R. Acad. Sci. Paris, Ser. I, 351
This paper deals with a two-competing-species chemotaxis system with two different chemicals { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ v t = Δ v − v + w , ( x , t ) ∈ Ω × ( 0 , ∞ ) , w t = Δ w − χ 2 ∇ ⋅ ( w ∇ z ) + μ 2 w ( 1 − a 2 u − w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ z t = Δ z − z + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n $\varOmega \subset \mathbb{R}^{n}$ ( n ≥ 1 ) $(n\geq 1)$ with the nonnegative initial data ( u 0 , τ v 0 , w 0 , τ z 0 ) ∈ C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) × C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) $(u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )$ , where τ ∈ { 0 , 1 } $\tau \in \{0,1\}$ and the parameters χ i , μ i , a i $\chi_{i},\mu_{i},a_{i}$ ( i = 1 , 2 $i=1,2$ ) are positive. When τ = 0 $\tau =0$ , based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if n = 2 $n=2$ . On the other hand, when τ = 1 $\tau =1$ , relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n ≥ 1 $n\geq 1$ and there exists θ 0 > 0 $\theta_{0}>0$ such that χ 2 μ 1 < θ 0 $\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}$ and χ 1 μ 2 < θ 0 $\frac{\chi_{1}}{\mu_{2}}<\theta_{0}$ .
Acta Applicandae Mathematicae – Springer Journals
Published: Nov 4, 2016
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