Long-Time Dynamics of the Cauchy Problem for the Predator–Prey Model with Cross-Diffusion

This paper is concerned with a predatorprey model in (Formula presented.) -dimensional spaces ((Formula presented.)), that describes the random movement of both predator and prey species, as well as the spatial dynamics involving predators pursuing prey and prey attempting to evade predators. The key findings are as follows: For (Formula presented.), any strong solutions converge to the heat kernel (Formula presented.) in (Formula presented.) -norm for all (Formula presented.) with the optimal decay rate (Formula presented.), revealing that diffusion ultimately dominates over cross-diffusion interactions. For (Formula presented.), suboptimal decay rates are obtained due to technical limitations of the bootstrap argument. This result clarifies the ultimate form of species diffusion in the entire space even with complex cross-diffusion, the population distribution still asymptotically approaches a Gaussian diffusion profile described by the heat kernel. At last, we also proved the existence of strong solutions for small initial data (indeed, even in the one-dimensional settings, only global weak solutions in a bounded domain had previously been successfully constructed), which forms the essential foundation for the preceding long-time dynamical analysis.