Attractors for singularly perturbed hyperbolic equations on unbounded domains

TL;DR

This paper studies the behavior of attractors for damped hyperbolic equations on unbounded domains as a parameter tends to zero, extending previous results to more general conditions including unbounded domains and critical growth nonlinearities.

Contribution

It extends the upper semicontinuity of attractors for hyperbolic equations to unbounded domains, critical nonlinearities, and less regular coefficients, without smoothness assumptions.

Findings

The family of attractors is upper semicontinuous as the parameter approaches zero.

The results apply to unbounded domains and nonlinearities with critical growth.

No smoothness assumptions are needed on domain boundary or coefficients.

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 $f(\cdot,u)$.