Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

Hongjun Qiu,Yinghui Zhang

doi:10.1155/2017/2708483

Abstract

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1≤p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

Highlights

In this paper, we consider the following 3D quasilinear hyperbolic system with nonlinear damping: Vt − div (h (V) P) = 0, x ∈ R3, t > 0, (1)Pt + ∇σ (V) = f (V) P, x ∈ R3, t > 0, with initial data (V, P) (x, 0) = (V0, P0) (x) 󳨀→ (V, 0), (2)as |x| 󳨀→ ∞, V > 0, where σ󸀠(V) < 0, h(V) > 0, f(V) < 0, and V > 0
We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K
If, Lp-norm (1 ≤ p < 6/5) of the initial perturbation is finite, we prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(V) given by Li and Saxton

Summary

Introduction

In [5], they obtained the convergence rates of the smooth solutions for the Cauchy problem under the following technical assumptions for the nonlinear damping: f󸀠󸀠 (V) < 0,. We use the special dissipation structure of (18)1-(18), the technique on interpolation and energy estimates, and a technical lemma on estimating the spatial derivatives of nonlinear function to tackle these difficulties. By virtue of the results obtained in the first two steps and by virtue of the uniform nonlinear energy estimates and the optimal Lp-L2 decay rates for the linearized system, we can prove the optimal Lp-L2 decay rates for the solutions.

Reformulation and the Proof of Theorem 1

LP-L2 Decay Estimates for the Linearized Equations

Uniform a Priori Estimates

Global Existence and Decay Rates