Dimension of attractors and invariant sets in reaction diffusion equations

Details Dimension of attractors and invariant sets in reaction diffusion equations Dimension of attractors and invariant sets in reaction diffusion equations

2011-02-20

arxiv.org/abs/1102.4062v1

Mathematics

Analysis of PDEs

20 pages

Scientific paper

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear reaction diffusion equation u_t+\beta(x)u-\Delta u&=f(x,u),&&(t,x)\in[0,+\infty[\times\Omega, u&=0,&&(t,x)\in[0,+\infty[\times\partial\Omega} {equation*} in $H^1_0(\Omega)$ 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 $\mathcal I$ needs not to be an 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.

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