Archive for Rational Mechanics and Analysis

, Volume 227, Issue 2, pp 715–747 | Cite as

Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species

  • Xiuqing Chen
  • Esther S. Daus
  • Ansgar JüngelEmail author
Open Access


The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager’s principle in thermodynamics. Under detailed balance (and without reaction) the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.

Mathematics Subject Classification

35K51 35Q92 92D25 60J10 



Open access funding provided by Austrian Science Fund (FWF).


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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of SciencesBeijing University of Posts and TelecommunicationsBeijingChina
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyWienAustria

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