General Decay of a Nonlinear Viscoelastic Wave Equation with Balakrishnân-Taylor Damping and a Delay Involving Variable Exponents

Abstract

This paper was aimed at investigating the stability of the following viscoelastic problem with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term $u_{tt}-\mathcal{M}\left({‖\nabla u‖}_2^2\right)\Delta u+\alpha \left(t\right){\int }_0^{t}g\left(t-s\right)\Delta u\left(s\right)\text{d}s+{\mu }_1{\left|u_t\right|}^{p\left(.\right)-2}{u}_t+{\mu }_2{\left|u_t\left(t-\tau \right)\right|}^{p\left(.\right)-2}{u}_t\left(t-\tau \right)=0\text{in}\Omega \times {ℝ}^{+},$ where $\Omega$ is a bounded domain of ${ℝ}^n$ , $p\left(.\right)\text{:}\overline{\Omega }⟶ℝ$ is a measurable function, $g>0$ is a memory kernel that decays exponentially, $\alpha \ge 0$ is the potential, and $\mathcal{M}\left({‖\nabla u‖}_2^2\right)=a+b{‖\nabla u\left(t\right)‖}_2^2+\sigma {\int }_{\Omega }\nabla u\nabla u_{t}dx$ for some constants $a>0$ , $b\ge 0$ , and $\sigma >0$ . Under some assumptions on the relaxation function, we use some suitable Lyapunov functionals to derive the general decay estimate for the energy. The problem considered is novel and meaningful because of the presence of the flutter panel equation and the spillover problem including memory and variable-exponent time delay control. Our result generalizes and improves previous conclusion in the literature.