Dimension of attractors and invariant sets in reaction diffusion equations
Keywords
Reaction diffusion equation, invariant set, attractor, dimensionAbstract
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.Downloads
Published
2012-04-23
How to Cite
1.
PRIZZI, Martino. Dimension of attractors and invariant sets in reaction diffusion equations. Topological Methods in Nonlinear Analysis. Online. 23 April 2012. Vol. 40, no. 2, pp. 315 - 336. [Accessed 28 March 2024].
Issue
Section
Articles
Stats
Number of views and downloads: 0
Number of citations: 0