Superlinear transmission in an indirect signal production chemotaxis system

Abstract

In this paper, the indirect signal production system with nonlinear transmission is considered $\left(\begin{array}{cc}& u_t=\Delta u-\nabla \cdot (u\nabla v),\\ & v_t=\Delta v-v+w,\\ & w_t=\Delta w-w+f\left(u\right)\end{array}\right)$in a bounded smooth domain $\Omega \subset R^n$ ($n\ge 1$) associated with homogeneous Neumann boundary conditions, where $f\in C^1\left([0,\infty )\right)$ satisfies $0\le f\left(s\right)\le s^{\alpha }$ with $\alpha >0$. It is known from [1] that the system possesses a global bounded solution if $0<\alpha <\frac{4}{n}$ when $n\ge 4$. In the case $n\le 3$ and if we consider superlinear transmission, no regularity of $w$ or $v$ can be derived directly. In this work, we show that if $0<\alpha <min\{\frac{4}{n},1+\frac{2}{n}\}$, the solution is global and bounded via an approach based on the maximal Sobolev regularity.