Global nonexistence of solutions for a system of viscoelastic wave equations with weak damping terms

Abstract

This paper deals with the initial boundary value problem for the viscoelastic wave equations$$\left\{\begin{array}{c}u_{t t}-\Delta u+\int_0^t g_1(t-\tau) \Delta u(\tau) d \tau+u_t=f_1(u, v), \\v_{t t}-\Delta v+\int_0^t g_2(t-\tau) \Delta v(\tau) d \tau+v_t=f_2(u, v)\end{array}\right.$$in a bounded domain. We obtain the global nonexistence of solutions by applying a lemma due to Y. Zhou [Global existence and nonexistence for a nonliear wave equation with damping and source terms, Math. Nacht, 278 (2005) 1341-1358].