Dimension of attractors and invariant sets of damped wave equations in unbounded domains
Keywords
Damped wave 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 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.Downloads
Published
2013-04-22
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1.
PRIZZI, Martino. Dimension of attractors and invariant sets of damped wave equations in unbounded domains. Topological Methods in Nonlinear Analysis. Online. 22 April 2013. Vol. 41, no. 2, pp. 267 - 285. [Accessed 28 March 2024].
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