On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains

M. Prizzi¹ ; K. P. Rybakowski²

¹ Università degli Studi di Trieste Dipartimento di Scienze Matematiche Via Valerio, 12/b 34100 Trieste, Italy
² Fachbereich Mathematik Universität Rostock Universitätsplatz 1 18055 Rostock, Germany

Studia Mathematica, Tome 154 (2003) no. 3, pp. 253-275

Résumé

We study a family of semilinear reaction-diffusion equations on spatial domains ${\mit \Omega }_\varepsilon $, $\varepsilon >0$, in ${\mathbb R}^ l $ lying close to a $k$-dimensional submanifold ${{\cal M}}$ of ${\mathbb R}^ l $. As $\varepsilon \to 0^+$, the domains collapse onto (a subset of) ${{\cal M}}$. As proved in [15], the above family has a limit equation, which is an abstract semilinear parabolic equation defined on a certain limit phase space denoted by $H^1_{\rm s}({\mit \Omega })$. The definition of $H^1_{\rm s}({\mit \Omega })$, given in the above paper, is very abstract. One of the objectives of this paper is to give more manageable characterizations of the limit phase space. Under additional hypotheses on the domains $ {\mit \Omega }_\varepsilon $ we also give a simple description of the limit equation. If, in addition, ${{\cal M}}$ is a $k$-sphere and the nonlinearity of the above equations is dissipative, then for every $\varepsilon >0$ small enough the corresponding equation on $ {\mit \Omega }_\varepsilon $ has an inertial manifold, i.e. an invariant manifold containing the attractor of the equation. We thus obtain the existence of inertial manifolds for reaction-diffusion equations on certain classes of thin domains of genuinely high dimension.

Détail

DOI : 10.4064/sm154-3-6

Keywords: study family semilinear reaction diffusion equations spatial domains mit omega varepsilon varepsilon mathbb lying close k dimensional submanifold cal mathbb varepsilon domains collapse subset cal proved above family has limit equation which abstract semilinear parabolic equation defined certain limit phase space denoted mit omega definition mit omega given above paper abstract objectives paper manageable characterizations limit phase space under additional hypotheses domains mit omega varepsilon simple description limit equation addition cal k sphere nonlinearity above equations dissipative every varepsilon small enough corresponding equation mit omega varepsilon has inertial manifold invariant manifold containing attractor equation obtain existence inertial manifolds reaction diffusion equations certain classes thin domains genuinely high dimension