### Continuity of attractors for net-shaped thin domains

#### Abstract

Consider a reaction-diffusion equation $u_t=\triangle u+f(u)$ on a family of net-shaped

thin domains $\Omega_\varepsilon$ converging to a one dimensional set as $\varepsilon\downarrow

0$. With suitable growth and dissipativeness conditions on $f$ these equations

define global semiflows which have attractors $\mathcal{A}_\varepsilon$. In [Th. Elsken, < i> A reaction-diffusion equation on a net-shaped thin domain< /i> , Studia Math.

< b> 165< /b> (2004), 159–199]

it has been shown that there is a limit problem which also defines a semiflow

having an attractor $\mathcal{A}_0$, and the family of attractors is

upper-semi-continuous at $\varepsilon=0$. Here we show that under a stronger

dissipativeness condition the family of attractors $\mathcal{A}_\varepsilon$,

$\varepsilon\ge 0$, is

actually continuous at $\varepsilon=0$.

thin domains $\Omega_\varepsilon$ converging to a one dimensional set as $\varepsilon\downarrow

0$. With suitable growth and dissipativeness conditions on $f$ these equations

define global semiflows which have attractors $\mathcal{A}_\varepsilon$. In [Th. Elsken, < i> A reaction-diffusion equation on a net-shaped thin domain< /i> , Studia Math.

< b> 165< /b> (2004), 159–199]

it has been shown that there is a limit problem which also defines a semiflow

having an attractor $\mathcal{A}_0$, and the family of attractors is

upper-semi-continuous at $\varepsilon=0$. Here we show that under a stronger

dissipativeness condition the family of attractors $\mathcal{A}_\varepsilon$,

$\varepsilon\ge 0$, is

actually continuous at $\varepsilon=0$.

#### Keywords

Reaction-diffusion equations; thin net shaped domain; continuity of attractors

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