A chemotaxis predator-prey model with indirect pursuit-evasion dynamics and parabolic signal (Q2661282)

The authors study predator-prey models with persuit evasion and nonlocal sensing, \[ \begin{aligned} & u_t-\Delta u+\nabla \cdot (u\nabla p) =\alpha w u-u\\ & w_t-D_w\Delta w-\nabla\cdot (w\nabla q)=\beta w (1-w-u)\\ & p_t-D_p\Delta p=\delta_ww-\delta_pp\\ & -D_q\Delta q=\delta_uu-\delta_qq \end{aligned} \tag{1} \] and \[ \begin{aligned} & u_t-\Delta u+\nabla \cdot (u\nabla p) =\alpha w u-u\\ & w_t-D_w\Delta w-\nabla\cdot (w\nabla q)=\beta w (1-w-u)\\ & \tau_1p_t-D_p\Delta p=\delta_ww-\delta_pp\\ & q_t-D_q\Delta q=\delta_uF(u)-\delta_qq \end{aligned}\tag{2} \] in \(\Omega \times (0,T)\), where \(\Omega\subset \mathbb{R}^2\) is a bounded domain with smooth boundary, and \(0\leq F=F(u)\) is a Lipschitz continuous function satisfying \(F(0)=0\) and \(F(s)\leq Cs^\xi\) with \(0<\xi<1\). The homogeneous Neumann condition is imposed on the boundary for each component, and suitable sets of initial values are provided. The existence of the global-in-time classical solution is shown with an estimate, which is uniform in \(t\) under the compatibility condition of the initial value.