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 $\Omega_\varepsilon$: \begin{equation} u_t=\Delta u+f(u), t > 0,\ (x,y)\in\Omega_\varepsilon\\ \partial _{\nu_\varepsilon} u=0, \qquad t > 0,\ (x,y)\in\partial\Omega_\varepsilon. \tag{$E_\varepsilon$} \end{equation} 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 $\varepsilon \to 0$.