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

Abstract

We consider in this work the nonlinearly damped semilinear Petrovsky equation with general nonlinear dissipation and source term
$$
\frac{\partial ^{2}u}{\partial t^{2}}+\Delta ^{2}u-\Delta u^{\prime
}+\left\vert u\right\vert ^{p-2}u+\alpha g\left( u^{\prime }\right) =\beta
f\left( u\right) \text{ in }\Omega \times \left[ 0,+\infty \right[
$$
where \(\Omega\) is open and bounded domain in \(\mathbb{R}^{n}\) with a smooth boundary \(\partial \Omega =\Gamma ,\alpha ,\beta >0.\) For the nonlinear continuous term \(f\left( u\right)\) and for \(g\) continuous, increasing, satisfying \(g(0)=0\), we prove the global existence of its solutions by means the Faedo-Galerkin procedure combined with the stable set method in \(H_{0}^{2}\left( \Omega \right) .\) Furthermore, we show that this solution blows up in finite time, when the energy is negative.