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2007 On the finite-time blow-up of a non-local parabolic equation describing chemotaxis
Nikos I. Kavallaris, Takashi Suzuki
Differential Integral Equations 20(3): 293-308 (2007).

Abstract

The non-local parabolic equation \[ v_t=\Delta v+\frac{\lambda e^v}{\int_\Omega e^v}\quad\mbox{in $\Omega\times (0,T)$} \] associated with Dirichlet boundary and initial conditions is considered here. This equation is a simplified version of the full chemotaxis system. Let $\lambda^*$ be such that the corresponding steady-state problem has no solutions for $\lambda>\lambda^*$, then it is expected that blow-up should occur in this case. In fact, for $\lambda>\lambda^*$ and any bounded domain $\Omega\subset {\bf R}^2$ it is proven, using Trudinger-Moser's inequality, that $\int_{\Omega} e^{v(x,t)}dx\to \infty$ as $t\to T_{max}\leq \infty.$ Moreover, in this case, some properties of the blow-up set are provided. For the two-dimensional radially symmetric problem, i.e. when $\Omega=B(0,1),$ where it is known that $\lambda^*=8\,\pi,$ we prove that $v$ blows up in finite time $T^* < \infty$ for $\lambda>8\,\pi$ and this blow-up occurs only at the origin $r=0$ (single-point blow-up, mass concentration at the origin).

Citation

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Nikos I. Kavallaris. Takashi Suzuki. "On the finite-time blow-up of a non-local parabolic equation describing chemotaxis." Differential Integral Equations 20 (3) 293 - 308, 2007.

Information

Published: 2007
First available in Project Euclid: 20 December 2012

zbMATH: 1212.35233
MathSciNet: MR2293987

Subjects:
Primary: 35K60
Secondary: 35B05 , 35Q80 , 92C17

Rights: Copyright © 2007 Khayyam Publishing, Inc.

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Vol.20 • No. 3 • 2007
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