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
HTML articles powered by AMS MathViewer

by James M. Greenberg and Wolfgang Alt PDF

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

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.

References

Wolfgang Alt, Models for mutual attraction and aggregation of motile individuals, Mathematics in biology and medicine (Bari, 1983) Lecture Notes in Biomath., vol. 57, Springer, Berlin, 1985, pp. 33–38. MR 812874, DOI 10.1007/978-3-642-93287-8_{4}
W. Alt, Degenerate diffusion equations with drift functionals modelling aggregation, Nonlinear Anal. 9 (1985), no. 8, 811–836. MR 799886, DOI 10.1016/0362-546X(85)90020-3
Wolfgang Alt and Douglas A. Lauffenburger, Transient behavior of a chemotaxis system modelling certain types of tissue inflammation, J. Math. Biol. 24 (1987), no. 6, 691–722. MR 880453, DOI 10.1007/BF00275511
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosci. 56 (1981), no. 3-4, 217–237. MR 632161, DOI 10.1016/0025-5564(81)90055-9

Cell aggregation and differentiation in dictyostelium

Model for chemotaxis

30

Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Toshitaka Nagai and Masayasu Mimura, Some nonlinear degenerate diffusion equations related to population dynamics, J. Math. Soc. Japan 35 (1983), no. 3, 539–562. MR 702772, DOI 10.2969/jmsj/03530539

Asymptotic behavior of the interfaces to a nonlinear degenerate diffusion equation in population dynamics

Mathematical models for cellular behavior

Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Renate Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc. 292 (1985), no. 2, 531–556. MR 808736, DOI 10.1090/S0002-9947-1985-0808736-1