Description

• Title:
  Attractors for singularly perturbed damped wave equations on unbounded domains
• Creator:
  Prizzi, Martino;
  Rybakowski, Krzysztof P.
• Description:
  For an arbitrary unbounded domain $\Omega\subset\mathbb R^3$ and for $\varepsilon> 0$, we consider the damped hyperbolic equations $$ \varepsilon u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}=f(x,u), \leqno{(\text{\rm H}_\varepsilon)} $$ 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 ${\Cal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\varepsilon\to0$, the family $({\mathcal A_\varepsilon})_{\varepsilon\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2008.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 32, No 1 (September 2008); 1 - 20;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2008.032/8775
• Subject:
  Attractors;
  singular perturbations;
  reaction-diffusion equations;
  damped wave equations;
  Primary 35B41;
  Secondary 35B25;
  37L05
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2008.032
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other