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

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 $\mathcal A_\varepsilon^{tr}$.