Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

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 3 -norm. Furthermore, if, additionally, -norm (1 ≤ < 6/5) of the initial perturbation is finite, we also prove the optimal -2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping (V) given by Li and Saxton.