Inertial manifolds on squeezed domains

Abstract

Let $\Omega$ be an arbitrary smooth bounded domain in ${\mathbb{R}}^2$ and $\epsilon >0$ be arbitrary. Squeeze $\Omega$ by the factor $\epsilon$ in the $y$-direction to obtain the squeezed domain ${\Omega }_{\epsilon }=\{(x,\epsilon y)\mid (x,y)\in \Omega \}$. In this paper we study the family of reaction-diffusion equations $$ \alignedat 2 u_t&=Δu+f(u),&\quad &t>0, (x,y)\inΩ_ε\partial_{ν_ε} u&=0,& & t>0, (x,y)\in\partialΩ_ε,\endalignedat\tag $E_ε$ $$ where $f$ is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as $\epsilon \to 0$, the equations $(E_{\epsilon })$ have a limiting equation which is an abstract semilinear parabolic equation defined on a closed linear subspace of $H^1(\Omega )$. We also proved that the family ${\Cal A}_ε$ of the corresponding attractors is upper semicontinuous at $\epsilon =0$. In this paper we prove that, if $\Omega$ satisfies some natural assumptions, then the limiting equation can be characterized as a reaction-diffusion equation on a finite topological graph. Moreover, there is a family $\Cal M_ε$ of inertial $C^1$-manifolds for $(E_{\epsilon })$, of some fixed finite dimension $\nu$, and, as $\epsilon \to 0$, the flow on $\Cal M_ε$ converges in the $C^1$-sense to the limit flow on $\Cal M_0$.

39 pages, 3 figures. To appear in "Jour. Dynam. Differerential Equations"