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

Abstract

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