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

Abstract

We deal with the following predator-prey model involving nonlinear indirect chemotaxis mechanism

$$ \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u+\xi \nabla \cdot (u \nabla w)+a_{1}u(1-u^{r_{1}-1}-b_{1}v), \ &\ \ x\in \Omega , \ t>0, \\ v_{t}=\Delta v-\chi \nabla \cdot (v \nabla w)+a_{2}v(1-v^{r_{2}-1}+b_{2}u), \ &\ \ x\in \Omega , \ t>0, \\ w_{t}=\Delta w-w+z^{\gamma }, \ &\ \ x\in \Omega , \ t>0, \\ 0=\Delta z-z+u^{\alpha }+v^{\beta }, \ &\ \ x\in \Omega , \ t>0 , \end{array}\displaystyle \right . $$

under homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega \subset \mathbb{R}^{n}$ ($n\geq 1$), where the parameters $\xi ,\chi ,a_{1},a_{2},b_{1},b_{2},\alpha ,\beta ,\gamma >0$. It has been shown that if $r_{1}>1$, $r_{2}>2$ and $\gamma (\alpha +\beta )<\frac{2}{n}$, then there exist some suitable initial data such that the system has a global classical solution $(u,v,w,z)$, which is bounded in $\Omega \times (0,\infty )$. Compared to the previous contributions, in this work, the boundedness criteria are only determined by the power exponents $r_{1}$, $r_{2}$, $\alpha$, $\beta$, $\gamma$ and spatial dimension $n$ instead of the coefficients of the system and the sizes of initial data.