Curved thin domains and parabolic equations

M. Prizzi ¹   ; M. Rinaldi ²   ; K. P. Rybakowski ³

¹ Dipartimento di Scienze Matematiche Università degli Studi di Trieste Via Valerio, 12/b 34100 Trieste, Italy
² DISCAFF Viale Ferrucci, 33 28100 Novara, Italy
³ Fachbereich Mathematik Universität Rostock Universitätsplatz 1 18055 Rostock, Germany

Studia Mathematica, Tome 151 (2002) no. 2, pp. 109-140

Résumé

Consider the family $$ \eqalign{ = {\mit\Delta} u + G(u),\ \quad t>0,\, x\in {\mit\Omega}_\varepsilon,\cr \partial_{\nu_\varepsilon}u= 0,\ \quad t>0,\, x\in \partial {\mit\Omega}_\varepsilon,} \tag*{$(E_\varepsilon)$}$$ 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 $\varepsilon\to0^+$.

Détail

DOI : 10.4064/sm151-2-2

Keywords: consider family eqalign mit delta quad mit omega varepsilon partial varepsilon quad partial mit omega varepsilon tag* varepsilon semilinear neumann boundary value problems where varepsilon small set mit omega varepsilon thin domain mathbb possibly holes which collapses varepsilon curved k dimensional submanifold mathbb dissipative equation varepsilon has global attractor nbsp mathcal varepsilon identify limit equation family varepsilon prove convergence trajectories establish upper semicontinuity result family mathcal varepsilon varepsilon