Some recent results on thin domain problems

Abstract

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 $\Omega_\varepsilon$: $$ \alignedat 2 &u_t=\Delta u+f(u),&\quad &t> 0,\ (x,y)\in\Omega_\varepsilon\\ &\partial _{\nu_\varepsilon} u=0,& & t> 0,\ (x,y)\in\partial\Omega_\varepsilon. \endalignedat\tag $\text{\rm E}_\varepsilon$ $$ 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 $\varvarepsilonilon\to 0$.