Reaction-diffusion problems in cell tissues. (English) Zbl 0879.35075
Summary: We consider reaction diffusion problems describing a reaction occurring in a planar domain (regarded as cell tissue). The diffusivity is assumed to be large except in the neighborhood of curves (regarded as membranes), around which it is assumed to be small. The subregions determined by the membranes and by the boundary of the domain (tissue wall) are regarded as cells. We assume that the tissue wall is a barrier through which no substance can pass. We prove that the dynamics is described by a system of ordinary differential equations that can be explicitly exhibited through the parameters defining the reaction diffusion problem, the length of the membranes, and the area of the cells. The tools employed are a detailed analysis of an eigenvalue problem and the invariant manifold theory.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 34D45 | Attractors of solutions to ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
Software:
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