Attractors for singularly perturbed hyperbolic equations on unbounded domains

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dc.creatorPrizzi, M.
dc.creatorRybakowski, K. P.
dc.date2007-03-21
dc.date.accessioned2026-07-07T07:53:06Z
dc.date.available2026-07-07T07:53:06Z
dc.descriptionFor an arbitrary unbounded domain $Ω\subset\R^3$ and for $\eps>0$, we consider the damped hyperbolic equations \leqno{(H_\eps)} \eps u_{tt}+ u_t+β(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}&=f(x,u),\quad x\in Ω, t\in\ro0,\infty.., u(x,t)&=0,\quad x\in \partial Ω, t\in\ro0,\infty... and their singular limit as $\eps\to0$, i.e. the parabolic equation \leqno{(P)} u_t+β(x)u- \sum_{ij}(a_{ij}(x)u_{x_j})_{x_i}&=f(x,u),\quad x\in Ω, t\in\ro0,\infty.., u(x,t)&=0,\quad x\in \partial Ω, t\in\ro0,\infty... Under suitable assumptions, $(H_\eps)$ possesses a compact global attractor $\Cal A_\eps$ in the phase space $H^1_0(Ω)\times L^2(Ω)$, while $(P)$ possesses a compact global attractor $\widetilde{\Cal A_0}$ in the phase space $H^1_0(Ω)$, which can be embedded into a compact set ${\Cal A_0}\subset H^1_0(Ω)\times L^2(Ω)$. We show that, as $\eps\to0$, the family $({\Cal A_\eps})_{\eps\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(Ω)\times H^{-1}(Ω)$. 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 $Ω$ be unbounded; last, we do not make any smoothness assumption on $\partialΩ$, $β(\cdot)$, $a_{ij}(\cdot)$ and $f(\cdot,u)$.
dc.description20 pages
dc.identifierhttps://arxiv.org/abs/math/0703642
dc.identifierhttp://arxiv.org/abs/math/0703642
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/126126
dc.subjectAnalysis of PDEs
dc.subjectDynamical Systems
dc.titleAttractors for singularly perturbed hyperbolic equations on unbounded domains
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