Determination of blowup type in the parabolic–parabolic Keller–Segel system

Abstract

This paper is concerned with a parabolic–parabolic Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \nabla \cdot ( \nabla u - u \nabla v ) &{} \quad \text{ in } \, \Omega \times (0,T), \\ v_t = \Delta v - \alpha v + u &{} \quad \text{ in } \, \Omega \times (0,T) \end{array} \right. \end{aligned}$$with a constant $$ \alpha \ge 0 $$ and nonnegative initial data in a smoothly bounded domain $$ \Omega \subset \mathbb {R}^2 $$ under the Neumann boundary condition or in $$ \Omega = \mathbb {R}^2 $$. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at $$ t = T 0 $$, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271:3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.