Calculus of Variations and Geometric Measure Theory

J. A. Carrillo - S. Fagioli - F. Santambrogio - M. Schmidtchen

Splitting schemes & segregation in reaction-(cross-)diffusion systems

created by santambro on 16 Nov 2017
modified on 26 Aug 2018


Accepted Paper

Inserted: 16 nov 2017
Last Updated: 26 aug 2018

Journal: SIAM J. Math. An.
Year: 2018


One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, \emph{i.e.} population densities with disjoint supports. We analyse such a reaction cross-diffusion system in 1D. In order to prove existence of weak solutions for a wide class of initial data without restriction about 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.