A note on the boundedness in a chemotaxis-growth system with nonlinear sensitivity

Abstract

This paper deals with a parabolic-elliptic chemotaxis-growth system with nonlinear sensitivity \begin{equation*}\label{1a} \begin{cases} u_t=\Delta u-\chi\nabla\cdot(\psi(u)\nabla v)+f(u), &(x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v-v+g(u), &(x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi>0$, the chemotactic sensitivity $\psi(u)\leq(u+1)^{q}$ with $q>0$, $g(u)\leq(u+1)^{l}$ with $l\in \mathbb{R}$ and $f(u)$ is a logistic source. The main goal of this paper is to extend a previous result on global boundedness by Zheng et al. [J. Math. Anal. Appl. 424(2015), 509–522] under the condition that $1\leq q+l<\frac{2}{n}+1$ to the case $q+l<1$.

Keywords

• boundedness
• chemotaxis-growth system
• chemotactic sensitivity