Inertial manifolds on squeezed domains

Let $Ω$ be an arbitrary smooth bounded domain in ² and $ε>0$ be arbitrary. Squeeze $Ω$ by the factor $ε$ in the y-direction to obtain the squeezed domain Ω_ε=\(x,εy) (x,y)∈Ω\. In this paper we study the family of reaction-diffusion equations 2 uₜ&=Δu+f(u),& &t>0, (x,y)∈Ω_ε∂ν_ε u&=0,& & t>0, (x,y)∈∂Ω_ε,E_ε$ $$ where f is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as ε→ 0, the equations (E_ε) have a limiting equation which is an abstract semilinear parabolic equation defined on a closed linear subspace of H¹(Ω). We also proved that the family A_ε of the corresponding attractors is upper semicontinuous at $ε=0$. In this paper we prove that, if $Ω$ 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 M_ε of inertial C¹-manifolds for (E_ε), of some fixed finite dimension $ν$, and, as ε→ 0, the flow on M_ε converges in the C¹-sense to the limit flow on M₀.