Description

• Title:
  Dimension of attractors and invariant sets of damped wave equations in unbounded domains
• Creator:
  Prizzi, Martino
• Description:
  Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation $$ \alignat 2 u_{tt}+\alpha u_t+\beta(x)u-\Delta u& =f(x,u), &\quad&(t,x)\in[0,+\infty\mathclose[\times\Omega, \\ u&=0,&\quad &(t,x)\in[0,+\infty\mathclose [\times\partial\Omega, \endalignat $$ in $H^1_0(\Omega)\times L^2(\Omega)$ has finite Hausdorff and fractal dimension. Here $\Omega$ is a regular, possibly unbounded, domain in $\mathbb{R}^3$ and $f(x,u)$ is a nonlinearity of critical 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 $f(x,u)$ is dissipative and $\mathcal I$ is the global attractor, we give an explicit bound on the Hausdorff and fractal dimension of $\mathcal I$ in terms of the structure parameters of the equation.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2013.04.22
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 41, No 2 (June 2013); 267 - 285;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2013.012/8318
• Subject:
  Damped wave equation;
  invariant set;
  attractor;
  dimension;
  35L70;
  35B41
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2013.012
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other