Description

• Title:
  Dimension of attractors and invariant sets in reaction diffusion equations
• Creator:
  Prizzi, Martino
• Description:
  Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation $$ \alignat 2 u_t+\beta(x)u-\Delta u&=f(x,u),&\quad &(t,x)\in[0,+\infty\mathclose[\times\Omega,\\ u&=0,&\quad &(t,x)\in[0,+\infty\mathclose\times\partial\Omega \endalignat $$ in $H^1_0(\Omega)$ has finite Hausdorff dimension. Here $\Omega$ is an arbitrary, possibly unbounded, domain in $\mathbb{R}^3$ and $f(x,u)$ is a nonlinearity of subcritical growth. The nonlinearity $f(x,u)$ needs not to satisfy any dissipativeness assumption and the invariant subset $\mathcal I$ needs not to be an attractor. If $\Omega$ is regular, $f(x,u)$ is dissipative and $\\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff dimension of $\mathcal I$ in terms of the structure parameter of the equation.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2012.04.23
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 40, No 2 (December 2012); 315 - 336;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2012.034/8382
• Subject:
  Reaction diffusion equation;
  invariant set;
  attractor;
  dimension;
  35L70;
  35B40;
  35B65
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2012.034
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other