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

Martino Prizzi

Abstract


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.

Keywords


Damped wave equation; invariant set; attractor; dimension

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