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2001 Chemotactic collapse in a parabolic-elliptic system of mathematical biology
Takasi Senba, Takashi Suzuki
Adv. Differential Equations 6(1): 21-50 (2001).

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.

Citation

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Takasi Senba. Takashi Suzuki. "Chemotactic collapse in a parabolic-elliptic system of mathematical biology." Adv. Differential Equations 6 (1) 21 - 50, 2001.

Information

Published: 2001
First available in Project Euclid: 2 January 2013

zbMATH: 0999.92005
MathSciNet: MR1799679

Subjects:
Primary: 92C17
Secondary: 35Q80, 92D15

Rights: Copyright © 2001 Khayyam Publishing, Inc.

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Vol.6 • No. 1 • 2001
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