Convergence rates of solutions in apredator-preysystem withindirect pursuit-evasion interaction in domains of arbitrary dimension

In this paper, we deal with the following indirect pursuit-evasion model $\begin{align} \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla w)+ u(\lambda-u+av), \quad x\in \Omega, t>0, \\ v_t v+\xi\nabla\cdot(v\nabla z)+ v(\mu-v-bu), { }{0 w- w+v}, z- z+ u}, \end{array}\right. \ (\star) \end{align} $ under homogeneous Neumann boundary conditions in a bounded domain \Omega\subset \mathbb{R}^N(N\geq1) smooth \partial\Omega $, where \chi, \xi, \lambda, \mu as well and b are positive parameters. This system is used to achieve some insight into possible dynamical properties of processes, which respective tactic movements oriented along gradients indirectly produced stimuli, rather than individuals directly. One main purpose present paper remove restriction N\leq3 $. Indeed, by using iteration argument combined suitable priori estimates, conclude that for any N\geq1 an associated initial-boundary value problem admits unique global classical solution. Moreover, large time behavior solutions also investigated. Specially speaking, when \chi<\left\{\begin{array}{ll} 4\sqrt{\frac{a(1+ab)}{b(\lambda+a\mu)}}, \quad\; \; \mbox{if}\; \lambda>b\mu, 4\sqrt{\frac{a}{b\lambda}}, \lambda\leq b\mu\ \xi<4\sqrt{\frac{b(1+ab)}{a(\mu-b\lambda)_+}} corresponding solution (u, v, w, z) decays (u_*, v_*, u_*) exponentially (or algebraically), u_* \frac{\lambda+a\mu}{1+ab}, v_* \frac{(\lambda-b\mu)_+}{1+ab} To best our knowledge, there first result on convergence rates system.