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Well-posedness, blowup, and global existence for an integrable shallow water equation
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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