Global stability in a multi-dimensional predator-prey system with prey-taxis

This paper studies the predator-prey systems with prey-taxis

$ \begin{eqnarray*} { \label{1.1}} \left\{ \begin{array}{llll} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\gamma uv-\rho u, \\ v_{t} = \Delta v-\xi uv+\mu v(1-v), \ \end{array} \right. \end{eqnarray*} $

in a bounded domain $ \Omega\subset\mathbb{R}^{n} $ $ (n = 2, 3) $ with Neumann boundary conditions, where the parameters $ \chi $, $ \gamma $, $ \rho $, $ \xi $ and $ \mu $ are positive. It is shown that the two-dimensional system possesses a unique global-bounded classical solution. Furthermore, we use some higher-order estimates to obtain the classical solutions with uniform-in-time bounded for suitably small initial data. Finally, we establish that the solution stabilizes towards the prey-only steady state $ (0, 1) $ if $ \rho>\gamma $ and towards the co-existence steady state $ (\frac{\mu(\gamma-\rho)}{\xi\rho}, \frac{\rho}{\gamma}) $ if $ \gamma>\rho $ under some conditions in the norm of $ L^{\infty}(\Omega) $ as $ t\rightarrow\infty $.