ASYMPTOTIC BEHAVIOR AND BLOW-UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATION WITH BOUNDARY DISSIPATION

Abstract

We study the nonlinear viscoelastic wave equation $$ u_{tt} - k_{0} \Delta u + \int_{0}^{t} g(t-s) \mathrm{div} \bigr[a(x) \nabla u(s) \bigr] ds + \bigr(k_{1} + b(x) |u_{t}|^{m-2}\bigr) u_{t} = |u|^{p-2}u $$ with dissipative boundary conditions. Under some restrictions on the initial data and the relaxation function and without imposing any restrictive assumption on $a(x)$, we show that the rate of decay is similar to that of $g$. We also prove the blow-up results for certain solutions in two cases. In the case $k_{1} = 0$, $m=2$, we show that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy. In another case, $k_{1}\geq 0$, $m\geq 2$, we prove a nonexistence result when the initial energy is less than potential well depth.