# [ W12 ]-ESTIMATES ON THE PREY-PREDATOR SYSTEMS WITH CROSS-DIFFUSIONS AND FUNCTIONAL RESPONSES

Shim, Seong-A

• Published : 2008.04.30
• 49 4

#### Abstract

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

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#### Cited by

1. On stability of two degenerate reaction–diffusion systems vol.390, pp.1, 2012, https://doi.org/10.1016/j.jmaa.2012.01.032
2. EXISTENCE OF GLOBAL SOLUTIONS FOR A PREY–PREDATOR MODEL WITH CROSS-DIFFUSION vol.03, pp.02, 2010, https://doi.org/10.1142/S1793524510000908