Superlinear transmission in an indirect signal production chemotaxis system

Abstract

In this paper, the indirect signal production system with nonlinear transmission is considered \[ \left\{ \begin{array}{lll} & u_t = Δu-\nabla\cdot(u \nabla v), \\ \displaystyle & v_t =Δv-v+w,\\ \displaystyle & w_t =Δw-w+ f(u) \end{array} \right. \] in a bounded smooth domain $\Omega \subset {\mathbb{R}}^n$ associated with homogenous Neumann boundary conditions, where $f\in C^1([0,\infty ))$ satisfies $0\le f(s)\le s^{\alpha }$ with $\alpha >0$. It is known 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.