New advances in some nonlinear anisotropic diffusion equations

Bruno Volzone (Politecnico di Milano)

Abstract: 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 self- similar 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. Vázquez.