Nour-Eddine Amroun, Abbes Benaissa

Global Existence and Energy Decay of Solutions to a Petrovsky Equation with General Nonlinear Dissipation and Source Term

Abstract:
We consider the nonlinearly damped semilinear Petrovsky equation
$$ u''-\Delta_{x}^{2}u+g(u')=b\ u|u|^{p-2}\quad \hbox{ on }\;\;\Omega\times [0, +\infty[ $$
and prove the global existence of its solutions by means of the stable set method in $H_{0}^{2}(\Omega)$ combined with the Faedo-Galerkin procedure. Furthermore, we study the asymptotic behavior of solutions when the nonlinear dissipative term $g$ does not necessarily
have a polynomial growth near the origin.

Keywords:
General nonlinear dissipation, nonlinear source, global existence, decay rate, multiplier method.

MSC 2000: 35L45, 93C20, 35B40, 35L70