Large time behavior of solutions for hyperbolic balance laws

Abstract

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 Lp-norm at a rate O(t−12(k+ρ+n2−np)) with p∈[2,∞] and ρ∈[0,n2] provided that initially ‖z0‖B˙2,∞−ρ<∞, where B˙2,∞−ρ is a homogeneous Besov space. These decay results do not impose an additional smallness assumption on Lp 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 ‖Pz0‖B˙2,∞−ρ+‖(I−P)z0‖B˙2,∞−ρ+1<∞ with ρ∈(n2,n+22], k-order derivatives of the solutions approach a constant state in the Lp-norm at a rate O(t−12(k+ρ+1+n2−np)).