Global existence and blow-up of a Petrovsky equation with general nonlinear dissipative and source terms

Abstract

"This work studies the initial boundary value problem for the Petrovsky equation with nonlinear damping \begin{equation*} \frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u-\Delta u^{\prime} +\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) =\beta f\left( u\right) \text{ in }\Omega \times \left[ 0,+\infty \right[, \end{equation*} where $\Omega $ is open and bounded domain in $\mathbb{R}^{n}$ with a smooth boundary $\partial \Omega =\Gamma$, $\alpha$, and $\beta >0$. For the nonlinear continuous term $f\left( u\right) $ and for $g$ continuous, increasing, satisfying $g$ $\left( 0\right) $ $=0$, under suitable conditions, the global existence of the solution is proved by using the Faedo-Galerkin argument combined with the stable set method in $H_{0}^{2}\left( \Omega \right)$. Furthermore, we show that this solution blows up in a finite time when the initial energy is negative."