Stability results for a diffusion equation with functional drift approximating a chemotaxis model

Greenberg, James M.; Alt, Wolfgang

doi:10.1090/S0002-9947-1987-0871674-4

Stability results for a diffusion equation with functional drift approximating a chemotaxis model
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by James M. Greenberg and Wolfgang Alt

Trans. Amer. Math. Soc. 300 (1987), 235-258

DOI: https://doi.org/10.1090/S0002-9947-1987-0871674-4

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Abstract:

A hyperbolic-parabolic "chemotaxis" system modelling aggregation of motile cells by production of a diffusible chemoattractant, is approximated by a scalar diffusion equation for the cell density, where the drift term is an explicit functional of the current density profile. We prove the unique existence and, using the Hopf-Cole transformation, the local stability of an equilibrium, i.e. a steady aggregation state. We also discuss the limiting hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps.

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