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.