Description

• Title:
  Some recent results on thin domain problems
• Creator:
  Prizzi, Martino;
  Rybakowski, Krzysztof P.
• Description:
  Let $\Omega$ be an arbitrary smooth bounded domain in $\mathbb R^2$ and $\varepsilon> 0$ be arbitrary. Write $(x,y)$ for a generic point of $\mathbb R^2$. Squeeze $\Omega$ by the factor $\varepsilon$ in the $y$-direction to obtain the squeezed domain $\Omega_\varepsilon=\{(x,\varepsilon y)\mid (x,y)\in\Omega\}$. Consider the following reaction-diffusion equation on $\Omega_\varepsilon$: $$ \alignedat 2 &u_t=\Delta u+f(u),&\quad &t> 0,\ (x,y)\in\Omega_\varepsilon\\ &\partial _{\nu_\varepsilon} u=0,&& t> 0,\ (x,y)\in\partial\Omega_\varepsilon. \endalignedat\tag $\text{\rm E}_\varepsilon$ $$ Here, $\nu_\varepsilon$ is the exterior normal vector field on $\partial \Omega_\varepsilon$ and $f\colon \mathbb R\to \mathbb R$ is a nonlinearity satisfying some growth and dissipativeness conditions ensuring that (E$_\varepsilon$) generates a semiflow $\pi_\varepsilon$ on $H^1(\Omega_\varepsilon)$ with a global attractor $\mathcal A_\varepsilon$. In this paper we report on some recent results concerning the asymptotic behavior of the equations (E$_\varepsilon$) as $\varvarepsilonilon\to 0$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  1999.12.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 14, No 2 (December 1999); 239 - 255;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.1999.031/10012
• Subject:
  General behaviour of solutions;
  intertial manifolds;
  35B10;
  35B42
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.1999.031
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other