On inertial manifolds for reaction-diffusion equations on genuinely high-dimensional thin domains
Volume 154 / 2003
Abstract
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.
