Nonlinear Diffusions (G. Grillo, M. Muratori, F. Punzo)

The role of equations of parabolic type is ubiquitous when modelling physical and biological systems. Indeed, the study of suitable classes of linear and nonlinear diffusion processes is still showing an outburst of new results and challenging problems. In the nonlinear setting, significant examples of diffusion phenomena are modelled by the porous media and the fast diffusion equations. The asymptotic behavior of singularly nonlinear differential equations is strongly dependent on the class of initial data under consideration. Detailed results have been given for initial data close to special explicit fundamental solutions, by means of entropy methods and establishing related functional inequalities, often with a clear geometrical meaning. Fine properties of solutions have been studied as well. In particular, propagation of positivity and local smoothing of singular equations have been tackled via techniques of functional-analytic flavor.

Having in mind the celebrated Li-Yau methods concerning Harnack inequalities and Hölder continuity properties of the heat equation on manifolds, we are concentrating on the role of the curvature in the behavior of solutions to nonlinear parabolic equations on manifolds, where appreciable similarities and sharp differences with respect to the flat case occur at the same time. In particular, we aim at dealing with asymptotics of general solutions, finite speed of propagation, existence and blow-up for large data, initial traces.

We also aim at studying fractional nonlinear diffusion in the Euclidean case, with particular emphasis on existence and uniquess for measure data, on the Cauchy problem for large data, on existence and uniqueness of distributional solutions, on the fine asymptotics of general solutions.

Politecnico di Milano - Dipartimento di Matematica