| Abstract: |
| In this talk, we focuses on the long-time dynamic behavior of the Cauchy problem related to a pursuit-evasion predator-prey model in $N$-dimensional spaces with $1\le N\le 3$. The system clearly adheres to the law of mass conservation, as evidenced by the fact that the $L^1$- norm remains constant. Our findings reveal that any global strong solution of this system converges to to the heat kernel in the sense of $L^p$-norm for any $1\le p\le \infty$. We also provide estimates on the decay rate of the solution, and obtain estimates on the decay rate of the solution that are consistent with those of the heat equation in $\mathbb R^N$ ($N=2, 3$), indicating their optimality. However, unfortunately, for one-dimensional case, despite our attempts to provide decay rate estimates, it is evident that this rate is not optimal. Additionally, as a supplementary result, we also verify the global existence and long-time asymptotic behavior of strong solutions for small initial values. |
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