2007/03/21 by Martino Prizzi, M. Prizzi, Prizzi, M. +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Stability and Controllability of Differential Equations #math.AP #math.DS
paper · pdf · doi:10.48550/arxiv.math/0703642
20 pages
arxiv created 2007/03/21 · openalex publication_date 2007/03/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an arbitrary unbounded domain Ω⊂\R3 and for \eps>0, we consider the damped hyperbolic equations \leqno(H_\eps) \eps utt+ ut+β(x)u- ∑ij(aij(x) uxj)xi&=f(x,u), x∈ Ω, t∈\ro0,∞.., u(x,t)&=0, x∈ ∂ Ω, t∈\ro0,∞... and their singular limit as \eps→0, i.e. the parabolic equation \leqno(P) ut+β(x)u- ∑ij(aij(x)uxj)xi&=f(x,u), x∈ Ω, t∈\ro0,∞.., u(x,t)&=0, x∈ ∂ Ω, t∈\ro0,∞... Under suitable assumptions, (H_\eps) possesses a compact global attractor \Cal A_\eps in the phase space H10(Ω)× L2(Ω), while (P) possesses a compact global attractor \widetilde\Cal A0 in the phase space H10(Ω), which can be embedded into a compact set \Cal A0⊂ H10(Ω)× L2(Ω). We show that, as \eps→0, the family (\Cal A_\eps)\eps∈[0,∞[ is upper semicontinuous with respect to the topology of H10(Ω)× 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 ∂Ω, β(⋅), aij(⋅) and f(⋅,u).