Superlinear transmission in an indirect signal production chemotaxis system

In this paper, the indirect signal production system with nonlinear transmission is considered $$\{\begin{array}{ll}& u_t=\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\mathrm{\nabla }v),\\ {\displaystyle }& v_t=\mathrm{\Delta }v-v+w,\\ {\displaystyle }& w_t=\mathrm{\Delta }w-w+f(u)\end{array}$$ in a bounded smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ associated with homogenous Neumann boundary conditions, where $f\in C^1([0,\mathrm{\infty }))$ 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.