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From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. (English) Zbl 1071.35001
In their seminal paper [J. Theor. Biol. 26, 399–415 (1970)], E. F. Keller and L. A. Segel introduced a class of quasilinear parabolic equations that proved to be of tremendous usefulness in various areas of the natural sciences, many of them totally unrelated to the phenomenon of slime mold aggregation the original model was supposed to describe and explain. In their simplest formulation, these equations, are of the form $u_t = \nabla (k_1(u,v)\nabla u -k_2 (u,v) \nabla v)\,$ $v_t = k_c \Delta v - k_3(v) v+uf(v),$ where the functions $$k_i$$ express the biological data of the problem, $$x \in \Omega \subset {\mathbb R}^n$$, $$t > 0$$, and the equations are equipped with Neumann boundary conditions and suitable initial data. It turns out that the mathematical object generated by these equations is very interesting and presents many challenging problems. Clearly, much depends on the choice of the functions $$k_i$$, different choices leading to different phenomena.
The present paper is the first part of a thorough survey of the work inspired by the Keller-Segel model. It contains 159 references and gives an up-to-date overview of a field in which, as the author says, “there still is an avalanche of publications running”.

The author gives a historical survey, reviewing the original work of Keller and Segel. He also presents macroscopic considerations relevant to the modelling of chemotaxis and explains the motivation for the reinforced random walk model of H. G. Othmer and A. Stevens [SIAM J. Appl. Math 57, 1044–1081 (1997; Zbl 0990.35128)]. He also details the contributions of Nanjundiah and Childress and Percus, which motivate much of subsequent work. The introduction is followed by sections on linear stability analysis of homogeneous states, existence and stability of non-homogeneous stationary solutions, characterisation of blowup (“chemotactic collapse”), post-blowup behaviour, existence of Liapunov functions and more general forms of the equations. The array of mathematical techniques brought to bear on the problem is impressive, encompassing “hard” functional analysis, energy and variational methods. The survey end with the presentation of the results of W. Alt (contained in his 1980 Habilitation) on comparison principles.

##### MSC:
 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 35K65 Degenerate parabolic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35J20 Variational methods for second-order elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35K50 Systems of parabolic equations, boundary value problems (MSC2000) 92C17 Cell movement (chemotaxis, etc.)
##### Keywords:
Keller-Segel chemotaxis model