Advances in Differential Equations

Chemotactic collapse in a parabolic-elliptic system of mathematical biology

Takasi Senba and Takashi Suzuki

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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

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

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92C17: Cell movement (chemotaxis, etc.)
Secondary: 35Q80: PDEs in connection with classical thermodynamics and heat transfer 92D15: Problems related to evolution


Senba, Takasi; Suzuki, Takashi. Chemotactic collapse in a parabolic-elliptic system of mathematical biology. Adv. Differential Equations 6 (2001), no. 1, 21--50.

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