Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4609272 | Journal of Differential Equations | 2016 | 36 Pages |
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≥3n≥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 LpLp-norm at a rate O(t−12(k+ρ+n2−np)) with p∈[2,∞]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 LpLp norm of the initial data and we thus improve the results in [3] and [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 LpLp-norm at a rate O(t−12(k+ρ+1+n2−np)).