Attractors for singularly perturbed damped wave equations on unbounded domains (Q1012994)

For an arbitrary unbounded domain \(\Omega\subset\mathbb R^3\) and for \(\varepsilon >0\), we consider the damped hyperbolic equations \[ \varepsilon u_{tt}+u_t +\beta (x)u - \sum_{ij} (a_{ij} (x)u_{x_j})_{x_i}=f(x,u), \] with Dirichlet boundary condition on \(\partial \Omega\), and their singular limit as \(\varepsilon\to 0\). Under suitable assumptions, the problem 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 \(H_0^1 (\Omega)\), which can be embedded into a compact set \(\mathcal{A}_0\subset H^1_0 (\Omega)\times L^2 (\Omega)\). We show that, as \(\varepsilon \to 0\), the family \((\mathcal{A}_\varepsilon)_{\varepsilon\in [0,\infty]}\) is upper semicontinuous with respect to the topology of \(H^1_0 (\Omega)\times H^{-1} (\Omega)\).