Dirac concentrations in Lotka-Volterra parabolic PDEs
| dc.creator | Perthame, Benoit | |
| dc.creator | Barles, Guy | |
| dc.date | 2007-08-28 | |
| dc.date.accessioned | 2026-07-07T08:26:05Z | |
| dc.date.available | 2026-07-07T08:26:05Z | |
| dc.description | We consider parabolic partial differential equations of Lotka-Volterra type, with a non-local nonlinear term. This models, at the population level, the darwinian evolution of a population; the Laplace term represents mutations and the nonlinear birth/death term represents competition leading to selection. Once rescaled with a small diffusion, we prove that the solutions converge to a moving Dirac mass. The velocity and weights cannot be obtained by a simple expression, e.g., an ordinary differential equation. We show that they are given by a constrained Hamilton-Jacobi equation. This extends several earlier results to the parabolic case and to general nonlinearities. Technical new ingredients are a $BV$ estimate in time on the non-local nonlinearity, a characterization of the concentration point (in a monomorphic situation) and, surprisingly, some counter-examples showing that jumps on the Dirac locations are indeed possible. | |
| dc.identifier | https://arxiv.org/abs/0708.3720 | |
| dc.identifier | http://arxiv.org/abs/0708.3720 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/136824 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 35B25, 35K57, 49L25, 92D15 | |
| dc.title | Dirac concentrations in Lotka-Volterra parabolic PDEs | |
| dc.type | text |