Abstract
In this talk, I will present results concerning a constructive approach to the Krein-Rutman theorem motivated by the asymptotic analysis of evolution PDEs. These results have been obtained in a recent work carried out in collaboration with C. Fonte and P. Gabriel. In this work, we develop, on the one hand, a general theory in an abstract framework that makes it possible to prove the existence of a " first eigentriplet " associated with the generator of a positive semi-group valid even in weakly dissipative cases where the first eigenvalue does not detach from the rest of the spectrum. This approach recovers and clarifies the classical theory. We also show results on the stability of the first eigenfunction and the return with constructive rate. We also illustrate our approach by applying it to several evolution PDEs. We will consider parabolic equations in different situations, including the case of sparsely regular coefficients in a bounded domain and the case of a sparsely confining field in all space. Our approach also applies to general transport models, growth-fragmentation equations, kinetic equations, kinetic Fokker-Planck type equations posed in a domain with reflection conditions, and selection-mutation models. In all these examples, we are able to generalize and/or refine previous results.
