Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0904.35037
Herrero, Miguel A.; Velázquez, Juan J.L.
A blow-up mechanism for a chemotaxis model. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No.4, 633-683 (1997). ISSN 0391-173X

Summary: We consider the following nonlinear system of parabolic equations: $$u_t= \Delta u-\chi\nabla(u\nabla v)\quad \Gamma v_t= \Delta v+ u- av\quad \text{for } x\in B_R,\quad t>0.\tag 1$$ Here $\Gamma$, $\chi$ and $a$ are positive constants, and $B_R$ is a ball of radius $R>0$ in $\bbfR^2$. At the boundary of $B_R$, we impose homogeneous Neumann conditions, namely: $${\partial u\over\partial n}= {\partial v\over\partial n}= 0\quad\text{for }| x|= R,\quad t> 0.\tag 2$$ 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.

MSC 2000:: *35K60 (Nonlinear) BVP for (non)linear parabolic equations
92C45 Kinetics in biochemical problems
35B40 Asymptotic behavior of solutions of PDE
35A20 Analytic methods (PDE)

Keywords: homogeneous Neumann conditions; Dirac-delta type singularity in finite time; inner layer around the unfolding singularity

Cited in: Zbl 1065.35066 Zbl 1060.92016 Zbl 1053.35064 Zbl 0984.35079

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]