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

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 $\mu(\cdot).$