Energy decay of a viscoelastic wave equation with supercritical nonlinearities

TL;DR

This paper investigates the long-term energy decay of solutions to a viscoelastic wave equation with supercritical nonlinearities, a fading memory term, and damping, establishing global existence and decay rates under certain conditions.

Contribution

It introduces a potential well framework for the system and proves global existence and energy decay for solutions with initial data in this well, considering supercritical source terms.

Findings

Established global existence of solutions within the potential well.

Derived uniform energy decay rates depending on the relaxation kernel and damping growth.

Extended previous work by including supercritical source terms and full past memory effects.

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 $-\infty$.