Chemotactic collapse in a parabolic system of mathematical biology

In 1970, Keller and Segel proposed a parabolic system describing the chemotactic feature of cellular slime molds and recently, several mathematical works have been devoted to it. In the present paper, we study its blowup mechamism and prove the following. First, chemotactic collapse occurs at each isolated blowup point. Next, any blowup point is isolated, provided that the Lyapunov function is bounded from below. Finally, only the origin can be a blowup point of radially symmetric solutions.

[1]  Paul Yang,et al.  Conformal deformation of metrics on $S^2$ , 1988 .

[2]  Takashi Suzuki,et al.  Concentration lemma, Brezis-Merle type inequality, and a parabolic system of chemotaxis , 2001 .

[3]  Hiroki Tanabe,et al.  Functional analytic methods for partial differential equations , 1996 .

[4]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[5]  Y. Giga.  A Local Characterization of Blowup Points of Semilinear Heat Equations , 1988 .

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

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

[8]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[9]  W. Jäger,et al.  On explosions of solutions to a system of partial differential equations modelling chemotaxis , 1992 .

[10]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

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

[12]  Takashi Suzuki,et al.  Chemotactic collapse in a parabolic-elliptic system of mathematical biology , 1999, Advances in Differential Equations.

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

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

[15]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[16]  Atsushi Yagi,et al.  NORM BEHAVIOR OF SOLUTIONS TO A PARABOLIC SYSTEM OF CHEMOTAXIS , 1997 .

[17]  H. Gajewski,et al.  Global Behaviour of a Reaction‐Diffusion System Modelling Chemotaxis , 1998 .

[18]  Ryo Ikehata,et al.  Stable and unstable sets for evolution equations of parabolic and hyperbolic type , 1996 .

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

[20]  Takashi Suzuki,et al.  LOCAL AND NORM BEHAVIOR OF BLOWUP SOLUTIONS TO A PARABOLIC SYSTEM OF CHEMOTAXIS , 2000 .

[21]  V. Nanjundiah,et al.  Chemotaxis, signal relaying and aggregation morphology. , 1973, Journal of theoretical biology.

[22]  Nicholas D. Alikakos,et al.  LP Bounds of solutions of reaction-diffusion equations , 1979 .