Radial solutions for elliptic equation ruled by Laplacian and pLaplacian
Using Fowler transformation and tecnhiques borrowed from non-autonomous dynamical systems theory (invariant manifold, Melnikov theory, ...) we can prove structure results for positive and nodal solutions for elliptic equation ruled by Laplacian and pLaplacian, with many different types of potentials
This techniques is particularly useful to study critical, supercritical and non-homogenous problems, to find singular solutions, to prove asymptotic estimates, and to find critical parameters and bifurcation phenomena.
Long time behavior of solutions of parabolic equations.
The parabolic reaction-diffusion equations with a (positive) power-like reaction term are a model for explosions and are therefore characterized by two typical behaviors: the initial condition (the temperature) is high enough to initate the explosion (and we have finite-time blow up) or it is too low and the temperature goes to 0.
We study the basin of attraction of the null solution and threshold phenomena, focusing in particular on singular data and in non-homogeneous contexts.
We combine maximum principle, comparison principle, sub and super solutions method with the knowledge of stationary radial ground states.
Melnikov theory for piecewise smooth systems.
Melnikov theory is a classical perturbative method useful to prove the persistence of homoclinic trajectories and the insurgence of caothic patterns. Our task is to extend the theory to the piecewise smooth case, assuming that the critical point may lie on the discontinuity surface (as in presence of dry friction) We focus in particular on phenomena which do not find a correspondence in the classical smooth setting.
Non autonomous bifurcation theory
We wish to extend to a non-autonomous setting bifurcations classically known in the autonomous case, such as saddle-node, transcritical, pitchfork and Hopf.
