Partial Differential Equations Seminar

Consider a scalar conservation law with a spatially discontinuous flux at a single point x = 0, and assume that the flux is uniformly convex when x ̸= 0. I will discuss controllability problems for AB-entropy solutions associated to the so-called (A, B)-interface connection. I will first present a characterization of the set of profiles of AB-entropy solutions at a time horizon T > 0, as fixed points of a backward-forward solution operator. Next, I will address the problem of identifying the set of initial data driven by the corresponding AB-entropy solution to a given target profile ω T, at a time horizon T > 0. These results rely on the introduction of proper concepts of AB-backward solution operator, and AB-genuine/interface characteristics associated to an (A, B)-interface connection, and exploit duality properties of backward/forward shocks for AB-entropy solutions.
 

Based on joint works with Luca Talamini (SISSA-ISAS, Trieste)

In this talk we describe several aspects related to the theory of some anisotropic parabolic equations. The anisotropy shown in such equations will appear in the form of porous medium, in the fast and porous medium diffusion regime. In particular, we show the existence of selfsimilar fundamental solutions, which is uniquely determined by its mass, and the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions. Time decay rates are derived as well as other properties of the solutions, like quantitative boundedness, positivity and regularity. 

The investigation of both models are objects of joint works with F. Feo and J. L. V´azquez.

TBA

TBA

TBA