Large time behavior of solutions for hyperbolic balance laws

We study the existence and the large time behavior of global solutions to the initial value problem for hyperbolic balance laws in n space dimensions with n >= 3 admitting an entropy and satisfying the stable condition. We first construct global existence of the solutions to such a system around a steady state if the initial energy is small enough. Then we show that k -order derivatives of these solutions approach a constant state in the L-P-norm at a rate 0(t (-1/2 (k+p+ n/2 -n/p))) with p is an element of [2, infinity] and rho is an element of [0, n/2] provided that initially vertical bar vertical bar z(0)vertical bar vertical bar (B) overdot(2,infinity)(-rho) where (B) overdot(2,infinity)(-rho) is is a homogeneous Besov space. These decay results do not impose an additional smallness assumption on L-P norm of the initial data and we thus improve the results in [3,19]. We also show faster decay results in the sense that if vertical bar vertical bar P-z0 vertical bar vertical bar (B) overdot(2,infinity)(-rho) + vertical bar vertical bar(I-P)z(0)vertical bar vertical bar (B) overdot(2,infinity)(-rho+1) < infinity with p is an element of (n/2, n+2/2] k -order derivatives of the solutions approach a constant state in the L-P -norm at a rate O(t(-1/2(k+rho+1+n/2-n/p))). (C) 2016 Elsevier Inc. All rights reserved.