Boundedness in the higher-dimensional quasilinear chemotaxis-growth system with indirect attractant production

This paper deals with the following quasilinear chemotaxis-growth system

in a smoothly bounded domain $\Omega ?{\mathbb{R}}^n(n\ge 3)$ under zero-flux boundary conditions. The parameters $\mu ,\delta$ and $\tau$ are positive and the diffusion function $D\left(u\right)$ is supposed to generalize the prototype $D\left(u\right)\ge D_0{u}^{\theta }$ with $D_0>0$ and $\theta \in \mathbb{R}$. Under the assumption $\theta >1?\frac{4}{n}$, it is proved that whenever $\mu >0$, $\tau >0$ and $\delta >0$, for any given nonnegative and suitably smooth initial data ($u_0$, $v_0$, $w_0$) satisfying $u_0?0$, the corresponding initial–boundary problem possesses a unique global solution which is uniformly-in-time bounded. The novelty of the paper is that we use the boundedness of the $\left|\right|v(?,t)|{|}_{W^{1,s}\left(\Omega \right)}$ with $s\in 1,\frac{2n}{n?2})$ to estimate the boundedness of $\left|\right|?v(?,t)|{|}_{L^{2q}\left(\Omega \right)}(q>1)$. Moreover, the result in this paper can be regarded as an extension of a previous consequence on global existence of solutions by Hu et al. (2016) under the condition that $D\left(u\right)\equiv 1$ and $n=3$.