Curved thin domains and parabolic equations
Volume 151 / 2002
Abstract
Consider the family $$ \eqalign{ &u_t = {\mit\Delta} u + G(u),\ \quad t>0,\, x\in {\mit\Omega}_\varepsilon,\cr &\partial_{\nu_\varepsilon}u= 0,\ \quad t>0,\, x\in \partial {\mit\Omega}_\varepsilon,} \tag*{$(E_\varepsilon)$}$$ of semilinear Neumann boundary value problems, where, for $\varepsilon>0$ small, the set ${\mit\Omega}_\varepsilon$ is a thin domain in $\mathbb R^l$, possibly with holes, which collapses, as $\varepsilon\to0^+$, onto a (curved) $k$-dimensional submanifold of $\mathbb R^l$. If $G$ is dissipative, then equation $(E_\varepsilon)$ has a global attractor ${\mathcal A}_\varepsilon$. We identify a “limit” equation for the family $(E_\varepsilon)$, prove convergence of trajectories and establish an upper semicontinuity result for the family ${\mathcal A}_\varepsilon$ as $\varepsilon\to0^+$.
