Some recent results on thin domain problems

Martino Prizzi, Krzysztof P. Rybakowski



Let $\Omega$ be an arbitrary smooth bounded domain in $\mathbb R^2$ and $\varepsilon> 0$ be
arbitrary. Write $(x,y)$ for a generic point of $\mathbb R^2$. Squeeze $\Omega$ by
the factor $\varepsilon$ in the
$y$-direction to obtain the squeezed domain $\Omega_\varepsilon=\{(x,\varepsilon y)\mid
(x,y)\in\Omega\}$. Consider the following reaction-diffusion equation on
$$ \alignedat 2
&u_t=\Delta u+f(u),&\quad &t> 0,\ (x,y)\in\Omega_\varepsilon\\
&\partial _{\nu_\varepsilon} u=0,&
& t> 0,\ (x,y)\in\partial\Omega_\varepsilon.
\endalignedat\tag $\text{\rm E}_\varepsilon$
Here, $\nu_\varepsilon$ is the exterior normal vector field on $\partial \Omega_\varepsilon$ and $f\colon
\mathbb R\to \mathbb R$ is a nonlinearity satisfying some growth and dissipativeness
conditions ensuring that (E$_\varepsilon$) generates a semiflow $\pi_\varepsilon$ on
$H^1(\Omega_\varepsilon)$ with a global attractor $\mathcal A_\varepsilon$.
In this paper we report on some recent results concerning the asymptotic
behavior of the equations (E$_\varepsilon$) as $\varvarepsilonilon\to 0$.


General behaviour of solutions; intertial manifolds

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