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

In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form

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 $b\vert u\vert^{p-1}u$ by $b\vert u\vert^{p-1}u -q(x)^2u$, where $q(x)$ is a sufficiently slowly decreasing function, an analogous result holds.