Splitting Schemes and Segregation in Reaction Cross-Diffusion Systems

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.