Attractors for singularly perturbed damped wave equations on unbounded domains

Abstract

For an arbitrary unbounded domain $\Omega\subset\mathbb R^3$ and for $\varepsilon> 0$, we consider the damped hyperbolic equations $$ \varepsilon u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}=f(x,u), \leqno{(\text{\rm H}_\varepsilon)} $$ with Dirichlet boundary condition on $\partial\Omega$, and their singular limit as $\varepsilon\to0$. Under suitable assumptions, (H$_\varepsilon)$ possesses a compact global attractor ${\mathcal A}_\varepsilon$ in $H^1_0(\Omega)\times L^2(\Omega)$, while the limiting parabolic equation possesses a compact global attractor $\widetilde{\mathcal A_0}$ in $H^1_0(\Omega)$, which can be embedded into a compact set ${\Cal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\varepsilon\to0$, the family $({\mathcal A_\varepsilon})_{\varepsilon\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$.