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

Mosbah Kaddour, Farid Messelmi


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.


global existence; blow-up; nonlinear dissipation; Petrovsky equation

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