Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

TL;DR

This paper investigates the blow-up behavior in a chemotaxis-growth system with spatially varying logistic terms, showing that blow-up points are confined to regions where the logistic damping coefficient vanishes.

Contribution

It characterizes the location of blow-up points in a fully parabolic chemotaxis system with spatial heterogeneity, linking blow-up to the zero set of the logistic damping coefficient.

Findings

Blow-up set is contained within the zeroes of (x).

Provides conditions under which blow-up occurs at specific locations.

Extends understanding of blow-up behavior in heterogeneous chemotaxis models.

Abstract

We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system \begin{align*} \begin{cases} u_t = \Delta u - \nabla\cdot(u\nabla v) + \kappa(x)u-\mu(x)u^2, \\ v_t = \Delta v - v + u \end{cases} \end{align*} in bounded domains $\Omega\subset \mathbb{R}^2$ and show that the blow-up set is contained in the set of zeroes of $\mu$.