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

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 $O((\log t)^2)$.