Differential and Integral Equations

On the finite-time blow-up of a non-local parabolic equation describing chemotaxis

Nikos I. Kavallaris and Takashi Suzuki

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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).

Article information

Source
Differential Integral Equations Volume 20, Number 3 (2007), 293-308.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356039503

Mathematical Reviews number (MathSciNet)
MR2293987

Zentralblatt MATH identifier
1212.35233

Subjects
Primary: 35K60: Nonlinear initial value problems for linear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35Q80: PDEs in connection with classical thermodynamics and heat transfer 92C17: Cell movement (chemotaxis, etc.)

Citation

Kavallaris, Nikos I.; Suzuki, Takashi. On the finite-time blow-up of a non-local parabolic equation describing chemotaxis. Differential Integral Equations 20 (2007), no. 3, 293--308. https://projecteuclid.org/euclid.die/1356039503.


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