Energy decay of a viscoelastic wave equation with supercritical nonlinearities

Abstract

This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: $$\begin{aligned} {\left\{ \begin{array}{ll} u_{tt}- k(0) \Delta u - \int \limits _0^{\infty } k'(s) \Delta u(t-s) \hbox {d}s +|u_t|^{m-1}u_t =|u|^{p-1}u, \text { in } \Omega \times (0,T), \\ u(x,t)=u_0(x,t), \quad \text { in } \Omega \times (-\infty ,0], \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a bounded domain in $$\mathbb R^3$$ with a Dirichlét boundary condition and $$u_0$$ represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified, provided that the history value $$u_0$$ is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel $$-k'(s)$$ as well as the growth rate of the damping term. This manuscript complements our previous work (Guo et al. in J Differ Equ 257:3778–3812, 2014, J Differ Equ 262:1956–1979, 2017) where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to $$-\infty $$ .