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The aim of this paper is to review the main recent results about the dynamics of nonlinear partial differential equations describing flux-saturated transport mechanisms, eventually in combination with porous media flow and/or reactions terms. The result is a system characterized by the presence of wave fronts which move defining an interface. This can be used to model different process in applications in a variety of areas as developmental biology or astrophysics. The concept of solution and its properties (well-posedness in a bounded variation scenario, Rankine-Hugoniot and geometric conditions for jumps, regularity results, finite speed of propagation, . . . ), qualitative study of these fronts (traveling waves in particular) and application in morphogenesis cover the panorama of this review.

This paper is intended to review recent results and open problems concerning the existence of steady states to the Maxwell-Schrödinger system. A combination of tools, proofs and results are presented in the framework of the concentration-compactness method.

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We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Optimal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and potential energies in terms of conserved quantities (mass and total energy) of the solutions of the VP system and a nonlinear stability result. Then we apply our estimates to the study of the large time asymptotics and observe two different regimes.