Georgian Mathematical Journal 13 (2006), No. 3, 397--410
Copyright Heldermann Verlag 2006
Global Existence and Energy Decay of Solutions to a Petrovsky Equation with General Nonlinear Dissipation and Source Term
Nour-Eddine Amroun
Djillali Liabès University, Faculty of Sciences, Dept. of Mathematics, B. P. 89, Sidi Bel Abbes 22000, Algeria
amroun_nour@yahoo.com
Abbes Benaissa
Djillali Liabès University, Faculty of Sciences, Dept. of Mathematics, B. P. 89, Sidi Bel Abbes 22000, Algeria
benaissa_abbes@yahoo.com
[Abstract-pdf]
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: 35L45, 93C20, 35B40, 35L70
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