Dimension of attractors and invariant sets in reaction diffusion equations

zbMATH Open1288.35103arXiv1102.4062MaRDI QIDQ388650

Martino Prizzi

Publication date: 3 January 2014

Published in: Topological Methods in Nonlinear Analysis (Search for Journal in Brave)

Abstract: Under fairly general assumptions, we prove that every compact invariant set $mathcalI$ of the semiflow generated by the semilinear reaction diffusion equation u_t+�eta(x)u-Delta u&=f(x,u),&&(t,x)in[0,+infty[ imesOmega, u&=0,&&(t,x)in[0,+infty[ imespartialOmega} {equation*} in $H_0^1(Omega)$ has finite Hausdorff dimension. Here $Omega$ is an arbitrary, possibly unbounded, domain in $R^3$ 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 $mathcalI$ needs not to be an an attractor. If $Omega$ is regular, $f(x,u)$ is dissipative and $mathcalI$ is the global attractor, we give an explicit bound on the Hausdorff dimension of $mathcalI$ in terms of the structure parameter of the equation.

Full work available at URL: https://arxiv.org/abs/1102.4062

zbMATH Keywords

Dirichlet boundary conditionCwickel-Lieb-Rozenblum inequality

Mathematics Subject Classification ID

Attractors (35B41) Reaction-diffusion equations (35K57) Initial-boundary value problems for second-order parabolic equations (35K20) Semilinear parabolic equations (35K58)

Related Items (2)

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