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 <TEX>$W^{1_2}$</TEX>-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 <TEX>$W^{1_2}$</TEX>-bound of the solutions and obtain the global existence in time.
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