On the asymptotic behaviour of solutions to the fractional porous medium   equation with variable density
  We are concerned with the long time behaviour of solutions to the fractional porous medium equation with a variable spatial density. We prove that if the density decays slowly at infinity, then the solution approaches the Barenblatt-type solution of a proper singular fractional problem. If, on the contrary, the density decays rapidly at infinity, we show that the minimal solution multiplied by a suitable power of the time variable converges to the minimal solution of a certain fractional sublinear elliptic equation. 
Gabriele Grillo, Matteo Muratori, Fabio Punzo
math.AP