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Some notes on periodic systems with linear part at resonance
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 
$\varepsilon\partial_t^2u+\gamma\partial_t u\Delta_x u+f(u)=g,\quad u_{\partial\Omega}=0,$
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 $\mathcal A_\varepsilon^{tr}$.
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