Publication: A blow-up mechanism for a chemotaxis model
Loading...
Full text at PDC
Publication Date
1997
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Scuola Normale Superiore
Citations
Exportar
Abstract
We consider the following nonlinear system of parabolic equations: (1) ut =Δu−χ∇(u∇v), Γvt =Δv+u−av for x∈B R, t>0. Here Γ,χ and a are positive constants and BR is a ball of radius R>0 in R2. At the boundary of BR, we impose homogeneous Neumann conditions, namely: (2) ∂u/∂n=∂v/∂n=0 for |x|=R, t>0.
Problem (1),(2) is a classical model to describe chemotaxis, i.e., the motion of organisms induced by high concentrations of a chemical that they secrete. In this paper we prove that there exist radial solutions of (1),(2) that develop a Dirac-delta type singularity in finite time, a feature known in the literature as chemotactic collapse. The asymptotics of such solutions near the formation of the singularity is described in detail, and particular attention is paid to the structure of the inner layer around the unfolding singularity.
Description
UCM subjects
Unesco subjects
Keywords
Citation
S. D. Eidelman, Parabolic systems, North-Holland, Amsterdam, 1969.
R. M. Ford - D. A. Lauffenburger, Analysis of chemotactical bacterial distribution in population migration assays using a mathematical model applicable to steep or shallow attractant gradients, Bull. Math. Biol. 53 (1991), 721–749.
E. F. Keller - L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol. 26 (1970), 399–415.
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc. 329 (1992), 819–824.
M. A. Herrero - J. J. L. Velázquez, Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996), 583–623.
M. A. Herrero - J. J. L. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35 (1996), 177–194.
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl. (1995), 1–21.