Classical and generalized solutions of an alarm-taxis model

In bounded, spatially two-dimensional domains, the system \begin{equation*} \left\lbrace\begin{alignedat}{3} u_t &= d_1 \Delta u && &&+ u(\lambda_1 - \mu_1 u - a_1 v - a_2 w), \\ v_t &= d_2 \Delta v &&- \xi \nabla \cdot (v \nabla u) &&+ v(\lambda_2 - \mu_2 v + b_1 u - a_3 w),\\ w_t &= d_3 \Delta w &&- \chi \nabla \cdot (w \nabla (uv)) &&+ w(\lambda_3 - \mu_3 w + b_2 u + b_3 v), \end{alignedat}\right. \end{equation*} complemented with initial and homogeneous Neumann boundary conditions, models the interaction between prey (with density $u$), predator (with density $v$) and superpredator (with density $w$), which preys on both other populations. Apart from random motion and prey-tactical behavior of the primary predator, the key aspect of this system is that the secondary predator reacts to alarm calls of the prey, issued by the latter whenever attacked by the primary predator. We first show in the pure alarm-taxis model, i.e. if $\xi = 0$, that global classical solutions exist. For the full model (with $\xi > 0$), the taxis terms and the presence of the term $-a_2 uw$ in the first equation apparently hinder certain bootstrap procedures, meaning that the available regularity information is rather limited. Nonetheless, we are able to obtain global generalized solutions. An important technical challenge is to guarantee strong convergence of (weighted) gradients of the first two solution components in order to conclude that approximate solutions converge to a generalized solution of the limit problem.