Sorting phenomena in a mathematical model for two mutually attracting/repelling species. (English) Zbl 1393.35248
The purpose of this paper is to study the interplay between (nonlinear) diffusion and nonlocal attractive/repulsive interactions. First a general survey of this interdisciplanary subject is given. The main issue is formula (13), the relation between one-dimensional steady states. It is proved that the supports of the two components of energy minimizers, a weak solution to (13), has zero Lebesgue measure. The stationary equation admits a unique solution. Finally, numerical simulations are illustrated by pictures. The proofs use Gâteaux derivative, Krein-Rutman theorem, implicit function theorem, Taylor formula.
Reviewer: Thomas Ernst (Uppsala)
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B36 | Pattern formations in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K65 | Degenerate parabolic equations |
| 35R09 | Integro-partial differential equations |
| 92D25 | Population dynamics (general) |
Keywords:
phase separation; spatial sorting; nonlinear diffusion; long-range attraction; stationary statesReferences:
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