Advances in Differential Equations

Grow-up rate of a radial solution for a parabolic-elliptic system in $\mathbf{R}^2$

Takasi Senba

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Abstract

We consider radial and positive solutions to a parabolic-elliptic system in $\mathbf{R}^2$. This system was introduced as a simplified version of the Keller-Segel model. The system has the critical value of the total mass. If the total mass of a solution is more than the critical value, the solution blows up in finite time. If the total mass of a solution is less than the critical value, the solution exists globally in time. Recently, some properties of solutions whose total mass is equal to the critical value have been investigated. In this paper, we construct a grow-up solution whose total mass is equal to the critical value. Furthermore, we show that the grow-up rate of the solution is equal to $O((\log t)^2)$.

Article information

Source
Adv. Differential Equations Volume 14, Number 11/12 (2009), 1155-1192.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854788

Mathematical Reviews number (MathSciNet)
MR2560872

Zentralblatt MATH identifier
1182.35054

Subjects
Primary: 35K55: Nonlinear parabolic equations 35K57: Reaction-diffusion equations 92C17: Cell movement (chemotaxis, etc.) 35B35: Stability 25B40

Citation

Senba, Takasi. Grow-up rate of a radial solution for a parabolic-elliptic system in $\mathbf{R}^2$. Adv. Differential Equations 14 (2009), no. 11/12, 1155--1192. https://projecteuclid.org/euclid.ade/1355854788.


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