Polynomial-exponential stability and blow-up solutions to a nonlinear damped viscoelastic Petrovsky equation

Abstract

This work is concerned with the initial boundary value problem for a nonlinear viscoelastic Petrovsky equation

$$\begin{aligned} u_{tt}+\Delta ^{2}u-\int _{0}^{t}g(t-\tau )\Delta ^{2}u(\tau )d\tau -\Delta u_{t}-\Delta u_{tt}+u_{t}|u_{t}|^{m-1}=u|u|^{p-1}. \end{aligned}$$

We prove that the solution energy has polynomial rate of decay, even if the kernel g decays exponentially provided $m>1$ while decay rates is exponentially in the case of weak dam**. The unbounded properties of solutions in two cases $m=1$ and $p>m\ge 1$ have been also investigated. For the first case, we prove the blow-up of solutions with different ranges of initial energy. For the second case, we prove blow-up of solutions under some restrictions on g when the initial energy is negative or non negative at less than potential well depth.