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In this paper we prove an uniqueness and stability theorem for the solutions of Cauchy problem for the systems $$\frac{\partial }{{\partial t}}u + \sum\limits_{i = 1}^n { \frac{\partial }{{\partial x_i }} } f^i (x,t,u) = g(x,t,u),$$ whereu is a vector function (u 1 (x, t),..., u r (x, t)),f i =(a 1 i (x, t, u),..., a r i (x, t, u)), i=1,...,n, g=(g 1 (x, t, u),...,g r (x, t, u),i G ℝ n and t≥0. We use the concept of entropy solution introduced by Kruskov and improved by Lax, Dafermos and others autors. We assume that the Jacobian matricesf u i are symmetric and the Hessian(a j i ) uu (i=1,...,n; j=1,...,r) are positive. We obtain uniqueness and stability inL loc 2 within the class of those entropy solutions which satisfy $$\frac{{u_j (---,x_i ,---,t)---u_j (---,y_i ,---,t)}}{{x_i - y_i }} \geqslant - K(t),$$ (i=1,...,n; j=1,...,r) for (−,x i ,−,t), (−,y i ,−,t) on a compact setD ⊂ ℝ n x (0, ∞) and a functionK(t)∈L loc 1 ([0, ∞)) depending onD. Here we denote by (−,x i ,−,t) and (−,y i ,−,t) two points whose coordinates only differ in thei-th space variable. At the end we relax the hypotheses of symmetry and convexity on the system and give a theorem of uniqueness and stability for entropy solutions which are locally Lipschitz continuous on a strip ℝ n x [0,T].
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Mar 4, 2007
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