Existence and uniform decay for a nonlinear viscoelastic equation with strong damping (Q2769911)

This paper is devoted to the existence and uniform decay rates of the energy of solutions for the nonlinear viscoelastic problem with strong damping NEWLINE\[NEWLINE\left\{ \begin{aligned} &|u_t|^\rho u_{tt}-\Delta u - \Delta u_{tt}+\int_0^tg(t-\tau)\Delta u(\tau) d\tau-\gamma\Delta u_t=0,\quad \text{in}\quad \Omega\times(0,\infty)\\ &u=0\quad \text{on} \quad\Gamma\times(0,\infty),\quad u(x,0)=u^0(x),\quad u_t(x,0)=u^1(x),\quad x\in\Omega, \end{aligned} \right.NEWLINE\]NEWLINE where \(\Omega\subset\mathbb R^n\) is a bounded domain with smooth boundary \(\Gamma\), and \(\rho\) is a real number such that \(0<\rho\leqslant 2/(n - 2)\) if \(n\geqslant 3\) or \(\rho>0\) if \(n = 1,2\), \(\gamma\geqslant 0\). Here, \(g\) represents the kernel of the memory term which is assumed to decay exponentially. In order to obtain the existence of global solutions to the problem, the Faedo-Galerkin method is used. By the perturbed energy method, the uniform decay rates of the energy are obtained assuming a strong damping \(\Delta u_t\) acting in \(\Omega\).