Inertial manifolds on squeezed domains

Let $\mathrm{\Omega }$ be an arbitrary smooth bounded domain in ${\mathbb{R}}^2$ and $ϵ>0$ be arbitrary. Squeeze $\mathrm{\Omega }$ by the factor $ϵ$ in the $y$-direction to obtain the squeezed domain $\Omega_\epsilon={(x,\epsilon y)\mid (x,y)\in\Omega }$\mathrm{.}Inthispaperwestudythefamilyofreaction-diffusionequations$$ \alignedat 2 u_t&=\Delta u+f(u),&\quad &t>0, (x,y)\in\Omega_\epsilon \partial_{\nu_\epsilon} u&=0,& & t>0, (x,y)\in\partial\Omega_\epsilon,\endalignedat\tag $E_{ϵ}$ $$where$f$ is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as $ϵ\to 0$, the equations $(E_{ϵ})$ have a limiting equation which is an abstract semilinear parabolic equation defined on a closed linear subspace of $H^1(\mathrm{\Omega })$. We also proved that the family ${\text{Cal}A}_{ϵ}$ of the corresponding attractors is upper semicontinuous at $ϵ=0$. In this paper we prove that, if $\mathrm{\Omega }$ 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 $\text{Cal}{M}_{ϵ}$ of inertial $C^1$-manifolds for $(E_{ϵ})$, of some fixed finite dimension $\nu$, and, as $ϵ\to 0$, the flow on $\text{Cal}{M}_{ϵ}$ converges in the $C^1$-sense to the limit flow on $\text{Cal}{M}_0$.