2025/04/16 by Jiaqi Liu, Yixuan Wang, Liu, Jiaqi +2
Mathematics · Biochemistry, Genetics and Molecular Biology · Medicine · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.2504.12231
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 -μρ2 (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any μ∈ [0,(1)/(3)). This range of μ is sharp; for μ≥ (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 L2 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 L2 space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation.