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

#### 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.

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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.