Asymptotic L-1-decay of solutions of the porous medium equations to self-similarity

We consider the flow of gas in an N-dimensional porous medium with initial density v(0)(x) greater than or equal to 0. The density v(x, t) then satisf ies the nonlinear degenerate parabolic equation v(t) = Delta v(m) where m > 1 is a physical constant. Assuming that integral(1 + \x\(2))v(0)(x) dx < i nfinity, we prove that v(x, t) behaves asymptotically, as t --> infinity, l ike the Barenblatt-Pattle solution V(\x\, t). We prove that the L-1-distanc e decays at a rate t(1/((N+2)m-N)). Moreover, if N = 1, we obtain an explic it time decay for the L-infinity-distance at a suboptimal rate. The method we use is based on recent results we obtained for the Fokker-Planck equatio n [2], [3].