Attractors for singularly perturbed hyperbolic equations on unbounded domains

Abstract

For an arbitrary unbounded domain $\Omega\subset\R^3$ and for $\eps>0$, we consider the damped hyperbolic equations \leqno{(H_\eps)} \eps u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u),\quad x\in \Omega, t\in\ro0,\infty.., u(x,t)&=0,\quad x\in \partial \Omega, t\in\ro0,\infty... and their singular limit as $\eps\to0$, i.e. the parabolic equation \leqno{(P)} u_t+\beta(x)u- \sum_{ij}(a_{ij}(x)u_{x_j})_{x_i}&=f(x,u),\quad x\in \Omega, t\in\ro0,\infty.., u(x,t)&=0,\quad x\in \partial \Omega, t\in\ro0,\infty... Under suitable assumptions, $(H_\eps)$ possesses a compact global attractor $\Cal A_\eps$ in the phase space $H^1_0(\Omega)\times L^2(\Omega)$, while $(P)$ possesses a compact global attractor $\widetilde{\Cal A_0}$ in the phase space $H^1_0(\Omega)$, which can be embedded into a compact set ${\Cal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\eps\to0$, the family $({\Cal A_\eps})_{\eps\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$. We thus extend a well known result by Hale and Raugel in three directions: first, we allow $f$ to have critical growth; second, we let $\Omega$ be unbounded; last, we do not make any smoothness assumption on $\partial\Omega$, $\beta(\cdot)$, $a_{ij}(\cdot)$ and $f(\cdot,u)$.