Global existence of solutions to Keller-Segel chemotaxis system with heterogeneous logistic source and nonlinear secretion

TL;DR

This paper proves the global existence of solutions to a Keller-Segel chemotaxis system with heterogeneous logistic sources and nonlinear secretion, under certain conditions on initial data and system functions.

Contribution

It establishes the first rigorous proof of global solutions for this complex chemotaxis model with spatially varying coefficients and nonlinear terms.

Findings

Global existence of solutions is proven under specific assumptions.

Conditions on initial data and functions ensure solution regularity.

The results extend understanding of chemotaxis models with heterogeneity.

Abstract

We study the following Keller-Segel chemotaxis system with logistic source and nonlinear secretion: \begin{align*} u_t=\Delta u- \nabla\cdot(u\nabla v)+\kappa(|x|)u-\mu(|x|)u^p\quad\text{and}\quad 0=\Delta v-v+u^\gamma, \end{align*} where $\kappa(\cdot),~\mu(\cdot):[0,R]\rightarrow [0,\infty)$, $\gamma\in (1,\infty)$, $p\in(\gamma+1,\infty)$ and $\Omega \subset \mathbb{R}^n, n\geq 2$. For this system, we prove the global existence of solutions under suitable assumptions on the initial condition and the functions $\kappa(\cdot)$ and $\mu(\cdot).$