2002/08/31 by Martino Prizzi, M. Prizzi, Prizzi, M. +3
Computer Science · Engineering · Mathematics · #35K57 #35K90 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations #math.AP #math.DS #msc:35K57 #msc:35K90
paper · pdf · doi:10.48550/arxiv.math/0209002
39 pages, 3 figures. To appear in "Jour. Dynam. Differerential Equations"
arxiv created 2002/08/31 · openalex publication_date 2002/08/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ω be an arbitrary smooth bounded domain in \R2 and ε>0 be arbitrary. Squeeze Ω by the factor ε in the y-direction to obtain the squeezed domain Ωε=\(x,εy)| (x,y)∈Ω\. In this paper we study the family of reaction-diffusion equations \alignedat 2 utamp;=Δu+f(u),amp; amp;tgt;0, (x,y)∈Ωε∂νε uamp;=0,amp; amp; tgt;0, (x,y)∈∂Ωε,\endalignedatEε where f is a dissipative nonlinearity of polynomial growth. In a previous paper we showed that, as ε→ 0, the equations (Eε) 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 \Cal Aε 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 \Cal Mε of inertial C1-manifolds for (Eε), of some fixed finite dimension ν, and, as ε→ 0, the flow on \Cal Mε converges in the C1-sense to the limit flow on \Cal M0.