ABSTRACT This project involves the study of certain Nonlinear Diffusion Equations and related Singular Elliptic and Linear Degenerate Parabolic problems. More precisely, the proposed work can be divided into the following projects: 1. Singular Parabolic equations of Fast and Super-Fast diffusion (in collaboration with M. Del Pino). 2. Semilinear Elliptic equations with Singular behavior (in collaboration with M. Del Pino). 3. Uniqueness of signed solutions to the Porous Medium equation - A continuity result for certain Degenerate Linear Parabolic equations (in collaboration with C. Kenig). The equations proposed have both physical and theoretical significance. For example, from the physical point of view, they arise as models for the dynamics of thin liquid films and also as models for the limiting density distribution in the kinetics of two gases obeying Boltzmann equation. Also, interest in such equations has long existed in other fields such as metallurgy and polymer science. From the mathematical point of view, they exhibit a qualitative behavior that has remarkable features, different than in previously studied cases. Their investigation leads to the development of new techniques of mathematical analysis. Another attractive feature of the proposed problems is their connection to Differential Geometry, as they arise in the so called Ricci Flow, which has recently gained remarkable attention.
