Gradient estimates and global existence of smooth solutions to a cross-diffusion system

Luan T. Hoang, Truyen V. Nguyen, Tuoc V. Phan

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26 Scopus citations

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 W1,p-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.

Original languageEnglish
Pages (from-to)2122-2177
Number of pages56
JournalSIAM Journal on Mathematical Analysis
Volume47
Issue number3
DOIs
StatePublished - 2015

Keywords

  • Calderón- Zygmund type estimates
  • Cross-diffusion system
  • Global existence
  • Global regularity
  • Gradient estimates
  • SKT system

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