Blow-up of solutions to the Keller-Segel model with tensorial flux in high dimensions

TL;DR

This paper investigates the Keller-Segel model with tensorial flux in high dimensions, providing evidence that finite-time blowup solutions can occur when the flux is represented by a constant matrix tensor.

Contribution

It demonstrates the existence of finite-time blowup solutions in dimensions three and higher for Keller-Segel models with tensorial flux of a specific form.

Findings

Finite-time blowup solutions exist in dimensions n≥3.

Tensorial flux modeled as A∇v with constant matrix A leads to blowup.

Results extend understanding of blowup phenomena in chemotaxis models.

Abstract

Over the course of the last decade, there has been a significant level of interest in the analysis of Keller-Segel models incorporating tensorial flux. Despite this interest, the question of whether finite-time blowup solutions exist remains a topic of ongoing research. Our study provides evidence that solutions of this nature are indeed possible in dimensions $n\geq3,$ when utilizing a tensorial flux expressed in the form of $A\nabla v$, where $A$ denotes a matrix with constant components