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arXiv · 1311.6828

Self-diffusion and cross-diffusion equations: $w^{1,p}$-estimates and global existence of smooth solutions

Luan T. Hoang·Tuoc V. Phan·Truyen V. Nguyen
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Abstract

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global $W^{1,p}$-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.

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Luan T. Hoang, Tuoc V. Phan, Truyen V. Nguyen. 2013-11-26. Self-diffusion and cross-diffusion equations: $w^{1,p}$-estimates and global existence of smooth solutions. https://arxiv.org/abs/1311.6828

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