Bulletin of the
Korean Mathematical Society BKMS

The prey-predator system with a single cross-diffusion pressure is known to possess a local solution with the maximal existence time $T\le \infty$. By obtaining the bounds of $W^1_2$-norms of the local solution independent of $T$ we establish the global existence of the solution. And the long-time behaviors of the global solution are analyzed when the diffusion rates $d_1$ and $d_2$ are sufficiently large.

Keywords: prey-predator system, single cross-diffusion, self-diffusion, calculus inequalities, $W^1_2$-bound, long-time behavior

MSC numbers: 35K55, 35B40