ATTRACTORS FOR SINGULARLY PERTURBED DAMPED WAVE EQUATIONS ON UNBOUNDED DOMAINS

For an arbitrary unbounded domain Omega subset of R-3 and for consider the damped hyperbolic equations (H-epsilon) epsilon u(tt) + u(t) +beta(x)u - Sigma(ij) (a(ij)(x)u(xj))x(i) = f(x,u), with Dirichlet boundary condition on partial derivative Omega, and their singular limit as epsilon -> 0. Under suitable assumptions, (H-epsilon) possesses a compact global attractor A(epsilon) in H-0(1)(Omega) x L-2(Omega). while the limiting parabolic equation possesses a compact global attractor (A) over tilde (0) in H-0(1)(Omega), which can be embedded into a compact set A(0) subset of H-0(1)(Omega) x L-2(Omega). We show that, as epsilon -> 0, the family (A(epsilon)) (epsilon) (epsilon[0,infinity]) is upper semicontinuous with respect to the topology of H-0(1)(Omega) x H-1 (Omega).