Advances in Differential Equations

Chemotactic collapse in a parabolic-elliptic system of mathematical biology

Takasi Senba and Takashi Suzuki

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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


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