Stable solutions and nonstandard diffusions: PDE questions arising in Mathematical Physics

The concept of diffusion is ubiquitous in the physical sciences. From the mathematical point of view, its study started in the early 19th century with the development of PDE theory, and has many connections to Physics, Probability, Geometry, and Functional Analysis. This project aims to answer several outstanding questions related to the mathematics of diffusion.The proposal is divided into two blocks. The first one corresponds to the study of stable solutions to reaction-diffusion PDE, and more precisely the classification of global stable solutions in the physical space (i.e., in 3D) for a general class of problems including the Allen-Cahn, the Alt-Phillips, or the thin Alt-Caffarelli equations. We will also investigate the same question for complex-valued solutions in 2D, which arises in the construction of travelling waves for the Gross-Pitaevskii equation. The second block corresponds to nonstandard diffusions. In particular, we will study the Boltzmann equation (a fundamental model in statistical mechanics), nonlocal diffusions (deeply related to Lvy processes and ""anomalous diffusions''), as well as the porous medium equation (a classical nonlinear PDE that arises in various physical models in which diffusion is ""slow''). The highly ambitious goals of the project are motivated by some recent results obtained by the PI in these areas.""