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Abstract: We consider parabolic partial differential equations of Lotka-Volterra type,with a non-local nonlinear term. This models, at the population level, thedarwinian evolution of a population; the Laplace term represents mutations andthe nonlinear birth-death term represents competition leading to selection.Once rescaled with a small diffusion, we prove that the solutions converge to amoving Dirac mass. The velocity and weights cannot be obtained by a simpleexpression, e.g., an ordinary differential equation. We show that they aregiven by a constrained Hamilton-Jacobi equation. This extends several earlierresults to the parabolic case and to general nonlinearities. Technical newingredients are a $BV$ estimate in time on the non-local nonlinearity, acharacterization of the concentration point in a monomorphic situation and,surprisingly, some counter-examples showing that jumps on the Dirac locationsare indeed possible.



Autor: Benoit Perthame LJLL, Guy Barles LMPT

Fuente: https://arxiv.org/







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