Topological Methods in Nonlinear Analysis

Dimension of attractors and invariant sets of damped wave equations in unbounded domains

Martino Prizzi

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Abstract

Under fairly general assumptions, we prove that every compact invariant set $\mathcal I$ of the semiflow generated by the semilinear damped wave equation $$ \begin{alignat}{2} u_{tt}+\alpha u_t+\beta(x)u-\Delta u& =f(x,u), &\quad&(t,x)\in[0,+\infty[\times\Omega, \\ u&=0,&\quad &(t,x)\in[0,+\infty[\times\partial\Omega, \end{alignat} $$ 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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 41, Number 2 (2013), 267-285.

Dates
First available in Project Euclid: 21 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461245478

Mathematical Reviews number (MathSciNet)
MR3114308

Zentralblatt MATH identifier
1292.35058

Citation

Prizzi, Martino. Dimension of attractors and invariant sets of damped wave equations in unbounded domains. Topol. Methods Nonlinear Anal. 41 (2013), no. 2, 267--285. https://projecteuclid.org/euclid.tmna/1461245478


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