Abstract:

In this paper, we consider the following prey-taxis model with nonlinear diffusion and indirect signal production

$\left\{ \begin{array}{l}{u_t} = \Delta {u^{{m_1}}} - \chi \nabla \cdot \left( {u\nabla w} \right),\\{w_t} = \Delta w - \mu w + \alpha v{F_0}\left( u \right),\\{v_t} = \Delta {v^{{m_2}}} + \lambda v\left( {1 - \frac{v}{k}} \right) - v{F_0}\left( u \right)\end{array} \right.$

in a bounded domain of $ {{\mathbb{R}}^{3}}$ withzero-flux boundary condition. It is shown that for any m₁>1, m₂>1, there exists a global bounded weak solution for any large initial datum. Based on the uniform boundedness property, we also studied the large time behavior of solutions, and the global asymptotically stability of the constant steady states are established. More precisely, we showed that when λ=0, α ≥ 0, the global weak solution converges to (ū₀, 0, 0) in the large time limit; when λ>0, α=0, the global weak solution converges to (ū₀, 0, 0) if λ < F₀(ū), and the global weak solution converges to $\left( {{{\bar u}_0}, 0, k\left( {1 - \frac{{{F_0}(\bar u)}}{\lambda }} \right)} \right) $ if λ > F₀(ū).

Key words: Prey-Taxis, Porous Medium Diffusion, Global Weak Solution, Uniform Boundedness, Stabilization