Universit√© de Versailles 

Title: Remarks on the convergence to equilibrium in some parabolic evolution equations

Abstract: We consider a few examples of parabolic evolution equations in the whole space, linear or nonlinear, for which one can show the existence of a positive equilibrium and then one may prove convergence of positive solutions to a multiple of the equilibrium. A first example will be a class of Fokker-Planck equations, and then we give another example of a nonlinear mutation selection model, known as replicator-mutator equation in evolutionary biology. These models involve a nonlocal mutation kernel and a confining fitness potential. We prove that the long time behaviour of the Cauchy problem is determined by the underlying principal eigenelement of the underlying linear operator. To do so, through a minimization problem under constraints, we prove first the existence of such an eigenlement. Then we analyze the linear evolution problem through the proof of the existence of a spectral gap, by taking advantage of the theory of strongly continuous semigroups of positive operators. We conclude with the analysis of the nonlinear problem.