Diffusion, cross-diffusion, and their spike-layer steady states. (English) Zbl 0917.35047

Ni, Wei-Ming

Notices Am. Math. Soc. 45, No. 1, 9-18 (1998).

This is an expository paper on the effect of nonhomogeneous distribution of reactant with the evolution described by a system of diffusion equations. One distinctive characteristic of such models is that at least one of the reactants is highly concentrated in small areas and thereby exhibits strikingly different patterns. The author introduces the current research of those highly concentrated solutions, also known as spike-layers. The existence of spike-layer steady states, the profile of spikers and the location of their peaks, as well as their stability and instability properties are investigated.

Reviewer: Ibrahim Aganović (Zagreb)

MSC:

35K50	Systems of parabolic equations, boundary value problems (MSC2000)
35K57	Reaction-diffusion equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

location of peaks; nonhomogeneous distribution of reactant; highly concentrated solutions; stability and instability properties