Stability of Some Mechanisms of Chemotactic Aggregation

In this paper, the stability of a blow-up mechanism for a particular Keller--Segel model that yields chemotactic aggregation is considered. It is shown using matched asymptotic methods that the considered blow-up mechanism is stable. More precisely, small perturbations of the initial data produce a small shifting on the blow-up time and the blow-up point but not modification in the blow-up mechanism.

[1]  H. Berg,et al.  Dynamics of formation of symmetrical patterns by chemotactic bacteria , 1995, Nature.

[2]  M. A. Herrero,et al.  Generic behaviour of one-dimensional blow up patterns , 1992 .

[3]  M. A. Herrero,et al.  A blow-up mechanism for a chemotaxis model , 1997 .

[4]  F. Alcântara,et al.  Signal propagation during aggregation in the slime mould Dictyostelium discoideum. , 1974, Journal of general microbiology.

[5]  Piotr Biler,et al.  LOCAL AND GLOBAL SOLVABILITY OF SOME PARABOLIC SYSTEMS MODELLING CHEMOTAXIS , 1998 .

[6]  Angela Stevens,et al.  A Model for Gliding and Aggregation of Myxobacteria , 1991 .

[7]  L. Segel,et al.  Initiation of slime mold aggregation viewed as an instability. , 1970, Journal of theoretical biology.

[8]  H. Othmer,et al.  Models of dispersal in biological systems , 1988, Journal of mathematical biology.

[9]  M. A. Herrero,et al.  Chemotactic collapse for the Keller-Segel model , 1996, Journal of mathematical biology.

[10]  John Tyler Bonner,et al.  The Cellular Slime Molds , 1967 .

[11]  Miguel A. Herrero,et al.  Finite-time aggregation into a single point in a reaction - diffusion system , 1997 .

[12]  Takashi Suzuki,et al.  Chemotactic collapse in a parabolic system of mathematical biology , 2000 .

[13]  Dirk Horstmann,et al.  The nonsymmetric case of the Keller-Segel model in chemotaxis: some recent results , 2001 .

[14]  Jerome Percus,et al.  Nonlinear aspects of chemotaxis , 1981 .

[15]  M. A. Herrero,et al.  On the melting of ice balls , 1997 .

[16]  Alberto Bressan,et al.  Stable blow-up patterns , 1992 .

[17]  Takashi Suzuki,et al.  Concentration Behavior of Blow-up Solutions for a Simplified System of Chemotaxis (Variational Problems and Related Topics) , 2001 .

[18]  M. Cohen,et al.  Wave propagation in the early stages of aggregation of cellular slime molds. , 1971, Journal of theoretical biology.

[19]  Stephen Childress,et al.  Chemotactic Collapse in Two Dimensions , 1984 .

[20]  M. A. Herrero.  Asymptotic Properties of Reaction-Diffusion Systems Modeling Chemotaxis , 2000 .

[21]  M. A. Herrero,et al.  Singularity patterns in a chemotaxis model , 1996 .

[22]  Andrew J. Bernoff,et al.  Stability of Self-similar Solutions for Van der Waals Driven Thin Film Rupture , 1999 .

[23]  Andrea L. Bertozzi,et al.  Stable and unstable singularities in the unforced Hele‐Shaw cell , 1996 .

[24]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .