Finite time blowup for Keller-Segel equation with logistic damping in three dimensions
数学专题报告
报告题目(Title)：Finite time blowup for Keller-Segel equation with logistic damping in three dimensions
报告人(Speaker)：周涛 （新加坡国立大学）
地点(Place)：腾讯会议：265 764 641， 密码：619189
时间(Time)：2025年9月24日 （周三）16:00-17:00
邀请人(Inviter)：袁迪凡
报告摘要
The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-\mu \rho^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $\mu \in \big[0,\frac{1}{3}\big)$. This range of $\mu$ is sharp; for $\mu \ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation. This talk is based on the joint work with Jiaqi Liu (USC) and Yixuan Wang (Caltech).
