I posted on arXiv several months ago a new preprint Nonlocal porous medium equation: Barenblatt profiles and other weak solutions written with P. Biler and G. Karch.
Abstract: A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles — the well-known solutions of the classical porous medium equation.
New preprints coming soon!!