Asymptotic behavior of solutions for hyperbolic p -system with spatiotemporal nonlinear damping

Source abstract

This paper is devoted to the asymptotic behavior of the solution to a hyperbolic p-system with spatiotemporal nonlinear damping like −γ(x,t)u−ω(x,t)uq−1u,0≤λ<1. For the general case where γ(x, t) is a function of x and t, by ingeniously applying the approximate Green function method, we further enhance the decay rates of the asymptotic profile. Additionally, for the special case γ is only a function of t, we select a new asymptotic profile by performing the time asymptotic expansion of the solution to the original equation. In these two different cases, we separately consider the convergence of solutions to asymptotic profiles via energy estimates under the a priori assumptions. The convergence results ultimately obtained in this paper incorporate previous findings and represent a certain improvement.