Asymptotic stability of solutions to quasilinear damped wave equations with variable sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents:
$$ u_{tt}-\operatorname{div}\bigl( |\nabla u|^{r(x)-2}\nabla u\bigr)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$
with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show 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.