Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

Abstract:

In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form \begin{equation*} u_{tt}+ a|u_t|^{m-1}u_t - \Delta u = b|u|^{p-1}u\qquad \quad \text {in}\ [0,\infty )\times R^n. \qquad \qquad \end{equation*} Here $a,b>0.$ We prove that if $p>m \ge 1,$ then for any $\lambda >0$ there are choices of initial data from the energy space with initial energy $\mathcal {E}(0)=\lambda ^2,$ such that the solution blows up in finite time. If we replace $b|u|^{p-1}u$ by $b|u|^{p-1}u -q(x)^2u$, where $q(x)$ is a sufficiently slowly decreasing function, an analogous result holds.