Communications of the
Korean Mathematical Society CKMS

As a mathematical model proposed to understand the behaviors of interacting species, cross-diffusion systems with functional responses of prey-predator type are considered. In order to obtain $W^1_2$-estimates of the solutions, we make use of several forms of calculus inequalities and embedding theorems. We consider the quasilinear parabolic systems with the cross-diffusion terms, and without the self-diffusion terms because of the simplicity of computations. As the main result we derive the uniform $W^1_2$-bound of the solutions and obtain the global existence in time.

Keywords: quasilinear parabolic systems, calculus inequalities, local existence, global existence, cross-diffusions, self-diffusions, Holling-type II functional responses, uniform bounds

MSC numbers: 35K55, 35B40