Asymptotic stability of Solutions to Quasilinear Damped Wave Equations with Variable Sources

Abstract

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents \[{u_{tt}} - div( {{{| {\nabla u} |}^{r(x) - 2}}\nabla u} ) + {| {{u_t}} |^{m(x) - 2}}{u_t} - \Delta {u_t} = {| u |^{q( x ) - 2}}u\] with homogenous Dirichlet initial boundary value condition. Some energy estimate and Komornik inequality are used to prove some uniform estimate of decay rates of the solution. And then, we prove that $u(x, t) = 0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.