GLOBAL NONEXISTENCE OF ARBITRARY INITIAL ENERGY SOLUTIONS OF VISCOELASTIC EQUATION WITH NONLOCAL BOUNDARY DAMPING

Abstract

In this paper, we consider the long time behavior of solutions of the initial value problem for the viscoelastic wave equation under boundary damping \begin{eqnarray*} u_{tt}-\Delta u+\int_0^t g(t-\tau)\text{div}(a(x)\nabla u(\tau))d\tau+u_t=0 &\text{in}\,\Omega\times(0,\infty). \end{eqnarray*} For the low initial energy case, which is the non-positive initial energy, based on concavity argument we prove the blow up result. As for the high initial energy case, we give out sufficient conditions of the initial datum such that the solution blows up in finite time.