Some recent results on thin domain problems

Let $\Omega$ be an arbitrary smooth bounded domain in $\mathbb R^2$ and $\varepsilon> 0$ bearbitrary. Write $(x,y)$ for a generic point of $\mathbb R^2$. Squeeze $\Omega$ bythe 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 dissipativenessconditions 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 asymptoticbehavior of the equations (E$_\varepsilon$) as $\varvarepsilonilon\to 0$.