Recent results on thin domain problems II

Abstract

In this paper we survey some recent results on parabolic equations on curved squeezed domains. More specifically, consider the family of semilinear Neumann boundary value problems $$ \alignedat2 u_t &= \Delta u + f(u), &\quad&t> 0,\ x\in \Omega_\varepsilon, \\ \partial_{\nu_\varepsilon}u&= 0, &\quad& t> 0,\ x\in \partial \Omega_\varepsilon \endaligned \leqno{(\text{\rm E}_\varepsilon)} $$ where, for $\varepsilon> 0$ small, the set $\Omega_\varepsilon$ is a thin domain in $\mathbb R^\ell$, possibly with holes, which collapses, as $\varepsilon\to0^+$, onto a (curved) $k$-dimensional submanifold $\mathcal M$ of $\mathbb R^\ell$. If $f$ is dissipative, then equation (E$_\varepsilon$) has a global attractor ${\mathcal A}_\varepsilon$. We identify a ``limit'' equation for the family (E$_\varepsilon$), establish an upper semicontinuity result for the family ${\mathcal A}_\varepsilon$ and prove an inertial manifold theorem in case $\mathcal M$ is a $k$-sphere.