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{align*} \begin{cases} u_{tt}- k(0) \Delta u - \int_0^{\infty} k'(s) \Delta u(t-s) ds +|u_t|^{m-1}u_t =|u|^{p-1}u, \quad \text{ in } \Omega \times (0,T), \\ u(x,t)=u_0(x,t), \quad \text{ in } \Omega \times (-\infty,0], \end{cases} \end{align*} where $\Omega$ is a bounded domain in $\mathbb R^3$ with a Dirichl\'et 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 $-\infty$.