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We establish the existence and uniqueness of fundamental solutions for the fractional porous medium equation introduced in \cite{PQRV1}. They are self-similar functions of the form $u(x,t)= t^{-\alpha} f(|x|\,t^{-\beta})$ 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. Among other interesting qualitative properties of the equation we prove an Aleksandrov reflection principle.

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