We study the well-posedness of the Cauchy problem for a fractional porous medium equation with a varying density ρ>0. We establish existence of weak energy solutions; uniqueness and nonuniqueness is studied as well, according to the behavior of ρ at infinity. © 2013 Elsevier B.V. All rights reserved

We construct a smooth Lyapunov pair for a continuous differential inclusion possessing a compact attractor. Our approach is based on the introduction of an intrinsic length functional, together with the corresponding length distance, defined starting from the polar cone of the multivalued vector field. This method allows a more geometrical insight into the subject and leads to a proof of the result which is, in our opinion, quite simple compared with the others available in the literature.

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We study the existence and uniqueness of bounded weak solutions to a fractional sublinear elliptic equation with a variable coefficient, in the whole space. Existence is investigated in connection to a certain fractional linear equation, whereas the proof of uniqueness relies on uniqueness of weak solutions to an associated fractional porous medium equation with variable density

We consider Hamiltonians associated to optimal control problems for affine systems on the torus. They are not coercive and are possibly unbounded from below in the direction of the drift of the system. The main assumption is the strong bracket generation condition on the vector fields. We first prove the existence of a critical value of the Hamiltonian by means of the ergodic approximation. Next we prove the existence of a possibly discontinuous viscosity solution to the critical equation. We show that the long-time behaviour of solutions to the evolutive Hamilton-Jacobi equation is described in terms of the critical constant and a critical solution. As in the classical weak KAM theory we find a fixed point of the Hamilton-Jacobi-Lax-Oleinik semigroup, although possibly discontinuous. Finally we apply the existence and properties of the critical value to the periodic homogenization of stationary and evolutive H-J equations.