Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis

In this work we consider a two-species predator-prey chemotaxis model

$ \left\{ \begin{array}{lll} u_t = d_1\Delta u+\chi_1\nabla\cdot(u\nabla v)+u(a_1-b_{11}u-b_{12}v), &x\in \Omega, t>0, \\[0.2cm] v_t = d_2\Delta v-\chi_2\nabla\cdot(v\nabla u)+v(-a_2+b_{21}u-b_{22}v-b_{23}w), & x\in \Omega, t>0, \\[0.2cm] w_t = d_3\Delta w-\chi_3\nabla\cdot(w\nabla v)+w(-a_3+b_{32}v-b_{33}w), & x\in \Omega, t>0 \\ \end{array}\right.(\ast) $

in a bounded domain with smooth boundary. We prove that if (1.7)-(1.13) hold, the model ($ \ast $) admits a global boundedness of classical solutions in any physically meaningful dimension. Moreover, we show that the global classical solutions $ (u,v,w) $ exponentially converges to constant stable steady state $ (u_\ast,v_\ast,w_\ast) $. Inspired by [5], we employ the special structure of ($ \ast $) and carefully balance the triple cross diffusion. Indeed, we introduced some functions and combined them in a way that allowed us to cancel the bad items.