Long time dynamics of the Cauchy problem for the predator-prey model with cross-diffusion

This paper is concerned with a predator-prey model in $N$-dimensional spaces ($N=1, 2, 3$), given by \begin{align*}\left\{\begin{aligned} &\frac{\partial u}{\partial t}=\Delta u-\chi\nabla\cdot(u\nabla v),\\ &\frac{\partial v}{\partial t}=\Delta v+\xi\nabla\cdot(v\nabla u), \end{aligned}\right. \end{align*} which describes random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. It is shown that any global strong solutions of the corresponding Cauchy problem converge to zero in the sense of $L^p$-norm for any $1<p\le \infty$, and also converge to the heat kernel with respect to $L^p$-norm for any $1\le p\le \infty$. In particular, the decay rate thereof is optimal in the sense that it is consistent with that of the heat equation in $\mathbb R^N$ ($N=2, 3$). Undoubtedly, the global existence of solutions appears to be among the most challenging topic in the analysis of this model. Indeed even in the one-dimensional setting, only global weak solutions in a bounded domain have been successfully constructed by far. Nevertheless, to provide a comprehensive understanding of the main results, we append the conclusion on the global existence and asymptotic behavior of strong solutions, although certain smallness conditions on the initial data are required.