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

"This work studies the initial boundary value problem for 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 $\mathbb{R}^{n}$ a smooth $\partial \Omega =\Gamma$, $\alpha$, $\beta >0$. For continuous term $f\left( $g$ continuous, increasing, satisfying $\left( 0\right) $=0$, under suitable conditions, global existence of solution proved by using Faedo-Galerkin argument combined stable set method $H_{0}^{2}\left( \right)$. Furthermore, we show that this blows up finite time when energy negative."