Abstract
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.
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Communicated by F. Otto
The first author acknowledges support from the National Natural Science Foundation of China, Grant 11471050. The last two authors acknowledge partial support from the Austrian Science Fund (FWF), Grants P22108, P24304, P30000, F65, and W1245.
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Chen, X., Daus, E.S. & Jüngel, A. Global Existence Analysis of Cross-Diffusion Population Systems for Multiple Species. Arch Rational Mech Anal 227, 715–747 (2018). https://doi.org/10.1007/s00205-017-1172-6
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DOI: https://doi.org/10.1007/s00205-017-1172-6