On viscoelastic wave equation with nonlinear boundary damping and source term (Q618974)

The author is concerned with the existence and uniform decay rates of solutions of the viscoelastic wave initial-boundary value problem \[ \begin{aligned} &u''-\Delta u+\int_0^th(t-\tau)\Delta u(\tau)d\tau=|u|^\rho u\;\text{in}\;\Omega\times (0,\infty),\\ &u=0\;\text{on}\;\Gamma_0\times (0,\infty),\\ &\frac{\partial u}{\partial \nu}-\int_0^t h(t-\tau)\frac{\partial u}{\partial \nu}(\tau)d\tau+g(u')=0\;\text{on}\;\Gamma_1\times (0,\infty),\\ &u(x,0)=u_0,\;u'(x,0)=u_1, \end{aligned} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^n (n\geq 1)\) with smooth boundary \(\Gamma=\Gamma_0\cup \Gamma_1,\;\Gamma_0\neq \emptyset,\;\Gamma_0\cap \Gamma_1=\emptyset\). He proves the existence of solutions and uniform decay rates without imposing any restrictive growth assumption on the damping term and weakening the usual assumptions on the relaxation function.