Blow-up for a Three Dimensional Keller-Segel Model with Consumption of Chemoattractant

TL;DR

This paper studies the conditions under which solutions to a three-dimensional Keller-Segel model with chemoattractant consumption blow up, extending previous results to bounded domains and providing new estimates and criteria.

Contribution

It introduces generalized blow-up criteria and higher-order estimates for the Keller-Segel model in bounded domains, advancing understanding of blow-up behavior in 3D.

Findings

Derived higher-order estimates for solutions.

Established blow-up criteria extending previous results.

Proved local non-degeneracy at blow-up points.

Abstract

We investigate blow-up properties for the initial-boundary value problem of a Keller-Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller-Segel model, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4,5] from the whole space $\mathbb{R}^3$ to the case of bounded smooth domain $\Omega\subset \mathbb{R}^3$. Lower global blow-up estimate on $\|n\|_{L^\infty(\Omega)}$ is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.