Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 50, Issue 5, Pages 5695-5718Publisher
SIAM PUBLICATIONS
DOI: 10.1137/17M1158379
Keywords
cross-diffusion; reaction diffusion; splitting schemes; variational schemes; segregation; pattern formation
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Funding
- Engineering and Physical Sciences Research Council (EPSRC) [EP/P031587/1]
- University of L'Aquila DP-LAND (Deterministic Particles for Local and Nonlocal Dynamics)
- Erasmus Mundus programme MathMods
- joint Imperial-CNRS Fellowship
- EPSRC [EP/P031587/1] Funding Source: UKRI
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One of the most fascinating phenomenon observed in reaction diffusion systems is the emergence of segregated solutions, i.e., population densities with disjoint supports. We analyze such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction of their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.
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