February & March  2004, 11(2&3): 351-392. doi: 10.3934/dcds.2004.11.351

Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities

1. 

Université de Poitiers, Laboratoire d'Applications des Mathématiques - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex

Received  November 2002 Revised  February 2004 Published  June 2004

We study the asymptotic behavior of weak energy solutions of the following damped hyperbolic equation in a bounded domain $\Omega\subset\R^3$:

$\varepsilon\partial_t^2u+\gamma\partial_t u-\Delta_x u+f(u)=g,\quad u|_{\partial\Omega}=0,$

where $\gamma$ is a positive constant and $\varepsilon>0$ is a small parameter. We do not make any growth restrictions on the nonlinearity $f$ and, consequently, we do not have the uniqueness of weak solutions for this problem.
We prove that the trajectory dynamical system acting on the space of all properly defined weak energy solutions of this equation possesses a global attractor $\mathcal A_\varepsilon^{tr}$ and verify that this attractor consists of global strong regular solutions, if $\varepsilon>0$ is small enough. Moreover, we also establish that, generically, any weak energy solution converges exponentially to the attractor $\mathcal A_\varepsilon^{tr}$.

Citation: Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351
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