Asymptotic and validity problems for Vlasov-type equations

In this thesis we study some asymptotic and validity problems concerning Vlasov-type equations. In the first part of the work we focus on the Landau damping and the long-time behavior of solutions of the Vlasov-HMF and Vlasov-Poisson equations. We do it by looking at the scattering problem, where the asymptotic datum is fixed, in the style of the work [17, Bibliog. Part I]. In the second part of the thesis we focus on validity problems for kinetic equations with topological interaction. This interaction does not depend on the metric distance but rather on the proximity rank among the agents and, in the last decade, has been widely used to describe biological systems that exhibit collective behaviors.