Type II blowup in the doubly parabolic Keller-Segel system in two dimensions
Type II blowup in the doubly parabolic Keller-Segel system in two dimensions
复制标题
二维双抛物线 Keller-Segel 系统中的 II 型爆炸
DOI：
10.1016/j.jfa.2016.09.016
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Noriko Mizoguchi
中科院分区：
文献类型：
--
作者：
H. Mitake;K. Soga;Seiichi Kamada;Noriko Mizoguchi
研究了抛物-抛物型Keller-Segel系统{ut=∇⋅(∇u−u∇v)在Ω×(0，T)中，vt=Δv−v+u在Ω×(0，T)中，∂u∂ν=∂v∂ν=0在∂Ω×(0，T)上，u(x，0)=u0(X)≥0，v(x，0)=v0(X)≥0，v(x，0)=v0(X)ΩΩ⊂R2中的细菌聚集模型。如果LIM sup t∞T→T‖u(T)‖L∞(Ω)=+∞，则称解(u，v)在t=T<；+LIM爆破。当(u，v)在t=T处爆破时，如果u(T)‖L∞(Ω)≤C(T−t)−1对t∈[0，T)具有一些常数C>；0，则称为类型I，否则称为类型II。尽管有建模的动机，但在过去的几十年里，抛物-抛物系统中是否经常发生有限时间井喷一直是开放的。最近，文献[18]，[19]，[15]证明了存在一大类径向初值使得相应的解在有限时间内爆破。然而，他们没有提供任何关于爆炸行为的信息。井喷速度是井喷机理研究中最受关注的问题之一。在本文中，我们证明了在径向情形下，抛物-抛物系统的每个爆破都是II型的。
This paper is concerned with a parabolic–parabolic Keller–Segel system {u t=∇⋅(∇ u− u∇ v) in Ω×(0, T), v t= Δ v− v+ u in Ω×(0, T),∂ u∂ ν=∂ v∂ ν= 0 on∂ Ω×(0, T), u (x, 0)= u 0 (x)≥ 0, v (x, 0)= v 0 (x)≥ 0 in Ω in a smoothly bounded domain Ω⊂ R 2. Keller and Segel proposed it 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<+∞ if lim sup t→ T‖ u (t)‖ L∞(Ω)=+∞. When (u, v) blows up at t= T, the blowup is called type I if‖ u (t)‖ L∞(Ω)≤ C (T− t)− 1 for t∈[0, T) with some constant C> 0, and type II otherwise. In spite of motivation of the modeling, it has remained open for the past decades whether or not finite-time blowup occurs frequently in the parabolic–parabolic system. Recently it was shown in [18],[19],[15] that there exists a large class of radial initial data such that the corresponding solutions blow up in finite time. However they did not give any information on blowup behavior. Blowup rate is one of the greatest concerns in investigation of blowup mechanism. In this paper, we show that each blowup is of type II in the parabolic–parabolic system in radial case.