Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

The occurrence of finite time blow-up phenomenon in the Keller–Segel (KS) model has always been a significant area of interest for mathematicians. Despite extensive research on the blow-up phenomenon in KS models with signal production, Understanding of this phenomenon in models with signal consumption mechanisms has been scarce.This paper marks a preliminary investigation into this unexplored field. In this study, we employ a backward self-similar solution to demonstrate that the finite time blowup indeed occurs within this model. More precisely, in one-dimensional space, finite time blowing up corresponding to the chemotactic collapse phenomenon (the formation of Dirac \(\delta \) $\delta$ -singularity ) happens; in high-dimensional space, the self-similar solution will blow up everywhere. Finally, we also consider the special cases where the diffusion coefficient of bacteria or oxygen is 0. For these cases, chemotactic collapse phenomenon occurs in both one-dimensional and two-dimensional spaces.