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 $\varepsilon=0$.