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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

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


Authors: James M. Greenberg and Wolfgang Alt
Journal: Trans. Amer. Math. Soc. 300 (1987), 235-258
MSC: Primary 35K55; Secondary 35B35, 35Q99, 92A05, 92A09
MathSciNet review: 871674
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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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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0871674-4
PII: S 0002-9947(1987)0871674-4
Keywords: Diffusion equation with functional, chemotaxis model, Hopf-Cole transformation, shock waves
Article copyright: © Copyright 1987 American Mathematical Society