Research lines:

We study some degenerate and singular parabolic equations that enter the modelling of (even non-Newtonian) fluids in porous media. Our aim is proving merely structural properties of the solutions, with a particular stress on Harnack's inequalities. We investigate how such results could be extended to Dirichlet forms of order  p  as well. Moreover, we continue the study of the dependence of solutions to elliptic problems on perturbations of the domain. In doing that, we want to use minimal regularity assumptions on the boundary. This will lead to interesting applications to evolution problems in variabile domains. Furthermore, we deal with the existence, uniqueness, and long time behavior of the solution of the reaction-diffusion problems related to the models of FitzHugh-Nagumo and Hodgkin-Huxley for the electical propagation in the hart tissue. Finally, well-posedness and regularity for elliptic and parabolic problems with unbounded coefficients are studied (such problems have applications in Probability theory and Financial Mathematics).