Topological Methods in Nonlinear Analysis

Attractors for singularly perturbed damped wave equations on unbounded domains

Martino Prizzi and Krzysztof P. Rybakowski

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Abstract

For an arbitrary unbounded domain $\Omega\subset\mathbb R^3$ and for $\varepsilon> 0$, we consider the damped hyperbolic equations \begin{equation} \varepsilon u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}=f(x,u), \tag{$(\text{\rm H}_\varepsilon)$} \end{equation} with Dirichlet boundary condition on $\partial\Omega$, and their singular limit as $\varepsilon\to0$. Under suitable assumptions, (H$_\varepsilon)$ possesses a compact global attractor $\mathcal A_\varepsilon$ in $H^1_0(\Omega)\times L^2(\Omega)$, while the limiting parabolic equation possesses a compact global attractor $\widetilde{\mathcal A_0}$ in $H^1_0(\Omega)$, which can be embedded into a compact set ${\mathcal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\varepsilon\to0$, the family $({\mathcal A_\varepsilon})_{\varepsilon\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 32, Number 1 (2008), 1-20.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150459

Mathematical Reviews number (MathSciNet)
MR2466799

Zentralblatt MATH identifier
1213.35054

Citation

Prizzi, Martino; Rybakowski, Krzysztof P. Attractors for singularly perturbed damped wave equations on unbounded domains. Topol. Methods Nonlinear Anal. 32 (2008), no. 1, 1--20. https://projecteuclid.org/euclid.tmna/1463150459


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