The authors investigate a viscoelastic initial-boundary value problem with dissipative boundary conditions: \[ \begin{aligned} &u_{tt}-k_0\Delta u+\int_0^t g(t-s)\text{div}[a(x)\nabla u(s)]\,ds+(k_1+b(x)u_t^{m-2})u_t =| u|^{p-2}n,\, x\in\Omega,\\ &u(x,t)=0,\, x\in \Gamma_0,\\ & k_0\frac{\partial u}{\partial \nu}-\int_0^t g(t-s)(a(x)\nabla u(s))\cdot\nu\,ds+h(u_t)=0,\, x\in \Gamma_1, \\ & u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\,(n\geq 1)\) with a smooth boundary \(\partial\Omega=\Gamma_0\cup\Gamma_1,\;\bar\Gamma_0\cap \bar\Gamma_1=\emptyset.\) Under some restrictions on the initial data and the relaxation function they show that the rate of decay is similar to that of \(g\). The blow-up results are proved for certain solutions in two cases.