Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type

We establish the existence, uniqueness and main properties of the fundamental solutions for the fractional porous medium equation introduced in [51]. They are self-similar functions of the form u(x, t) = t(-alpha) f (vertical bar x vertical bar t(-beta)) with suitable alpha and beta. As a main application of this construction, we prove that the asymptotic behaviour of general solutions is represented by such special solutions. Very singular solutions are also constructed. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection principle.