  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 $f(\cdot,u)$.
