@article{a9bd73bc51c54953a40e859788fa7924,

title = "A non-standard evolution problem arising in population genetics",

abstract = "We study the evolution of the probability density of an a sexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.",

keywords = "vortex, reflection, boundary-coupled, degenerate, weak, equations, parabolic, dynamics, sheets, Gene, solution, symmetry, fixation, evolutionary, Gene fixation , evolutionary dynamics, degenerate parabolic equations, boundarycoupled weak solutions",

author = "Chalub, {F{\'a}bio Augusto da Costa Carvalho} and Souza, {Max O.}",

note = "The authors want to thank Peter Markowich for helpful comments, and an anonymous referee for pointing out a mistake in an earlier version of Proposition 4 and for general comments that improved the presentation. FACCC is partially supported by FCT/Portugal, grants POCI/MAT/57546/2004, PTDC/MAT/68615/2006 and PTDC/MAT/66426/2006. MOS is partially supported by FAPERJ/Brazil grants 170.382/2006 and 110.174/2009. MOS also thanks the support and hospitality of FCT/UNL and Complexo Interdisciplinar/UL. Part of this work has been done during the Special Semester on Quantitative Biology Analyzed by Mathematics, organized by RICAM, Austrian Academy of Sciences.",

year = "2009",

language = "English",

volume = "7",

pages = "489--502",

journal = "Communications In Mathematical Sciences",

issn = "1539-6746",

publisher = "International Press",

number = "2",

}