Global Classical Solutions to a Predator-Prey Model with Nonlinear Indirect Chemotaxis Mechanism

We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism $\{\begin{array}{ll}{u}_t=\mathrm{\Delta }u+\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+a_{1}u(1-u^{r_1-1}-b_{1}v),& x\in \mathrm{\Omega },t>0,\\ v_t=\mathrm{\Delta }v-\chi \mathrm{\nabla }\cdot (v\mathrm{\nabla }w)+a_{2}v(1-v^{r_2-1}+b_{2}u),& x\in \mathrm{\Omega },t>0,\\ w_t=\mathrm{\Delta }w-w+z^{\gamma },& x\in \mathrm{\Omega },t>0,\\ 0=\mathrm{\Delta }z-z+u^{\alpha }+v^{\beta },& x\in \mathrm{\Omega },t>0,\end{array}$ \( \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u+\xi \nabla \cdot (u \nabla w)+a_{1}u(1-u^{r_{1}-1}-b_{1}v), \ &\ \ x\in \Omega , \ t>0, \\ v_{t}=\Delta v-\chi \nabla \cdot (v \nabla w)+a_{2}v(1-v^{r_{2}-1}+b_{2}u), \ &\ \ x\in \Omega , \ t>0, \\ w_{t}=\Delta w-w+z^{\gamma }, \ &\ \ x\in \Omega , \ t>0, \\ 0=\Delta z-z+u^{\alpha }+v^{\beta }, \ &\ \ x\in \Omega , \ t>0 , \end{array}\displaystyle \right . \) under homogeneous Neumann boundary conditions in a bounded and smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ $\Omega \subset \mathbb{R}^{n}$ ( $n\ge 1$ $n\geq 1$ ), where the parameters $\xi ,\chi ,a_1,a_2,b_1,b_2,\alpha ,\beta ,\gamma >0$ $\xi ,\chi ,a_{1},a_{2},b_{1},b_{2},\alpha ,\beta ,\gamma >0$ . It has been shown that if $r_1>1$ $r_{1}>1$ , $r_2>2$ $r_{2}>2$ and $\gamma (\alpha +\beta )<\frac{2}{n}$ $\gamma (\alpha +\beta )<\frac{2}{n}$ , then there exist some suitable initial data such that the system has a global classical solution $(u,v,w,z)$ $(u,v,w,z)$ , which is bounded in $\mathrm{\Omega }\times (0,\mathrm{\infty })$ $\Omega \times (0,\infty )$ . Compared to the previous contributions, in this work, the boundedness criteria are only determined by the power exponents $r_1$ $r_{1}$ , $r_2$ $r_{2}$ , $\alpha$ $\alpha $ , $\beta$ $\beta $ , $\gamma$ $\gamma $ and spatial dimension $n$ $n$ instead of the coefficients of the system and the sizes of initial data.