## Advances in Differential Equations

- Adv. Differential Equations
- Volume 6, Number 1 (2001), 21-50.

### Chemotactic collapse in a parabolic-elliptic system of mathematical biology

Takasi Senba and Takashi Suzuki

#### Abstract

We study the blowup mechanism for a simplified system of chemotaxis. First, Moser's iteration scheme is applied and the blowup point of the solution is characterized by the behavior of the local Zygmund norm. Then, Gagliardo-Nirenberg's inequality gives $\varepsilon_0>0$ satisfying $\limsup_{t\uparrow T_{\max}}\left\Vert u(t)\right\Vert_{L^1\left(B_R(x_0)\cap \Omega\right)}\geq \varepsilon_0$ for any blowup point $x_0\in \overline{\Omega}$ and $R>0$. On the other hand, from the study of the Green's function it appears that $t\mapsto\left\Vert u(t)\right\Vert_{L^1\left(B_R(x_0)\cap\Omega\right)}$ has a bounded variation. Those facts imply the finiteness of blowup points, and then, the chemotactic collapse at each blowup point and an estimate of the number of blowup points follow.

#### Article information

**Source**

Adv. Differential Equations Volume 6, Number 1 (2001), 21-50.

**Dates**

First available in Project Euclid: 2 January 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1357141500

**Mathematical Reviews number (MathSciNet)**

MR1799679

**Zentralblatt MATH identifier**

0999.92005

**Subjects**

Primary: 92C17: Cell movement (chemotaxis, etc.)

Secondary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer 92D15: Problems related to evolution

#### Citation

Senba, Takasi; Suzuki, Takashi. Chemotactic collapse in a parabolic-elliptic system of mathematical biology. Adv. Differential Equations 6 (2001), no. 1, 21--50. https://projecteuclid.org/euclid.ade/1357141500