### Dimension of attractors and invariant sets in reaction diffusion equations

#### Abstract

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.

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.

#### Keywords

Reaction diffusion equation; invariant set; attractor; dimension

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