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

Abstract

We consider the flow of gas in an N-dimensional porous medium with initial density v 0 (x) 0. The density v(x, t) then satisfies the nonlinear degenerate parabolic equation v t = v m where m > 1 is a physical constant. Assuming that (1 + |x| 2 )v 0 (x) dx < , we prove that v(x, t) behaves asymptotically, as t , like the Barenblatt-Pattle solution V (|x|, t). We prove that the L 1 -distance decays at a rate t 1/((N+2)m-N) . Moreover, if N = 1, we obtain an explicit time decay for the L -distance at a suboptimal rate. The method we use is based on recent results we obtained for the Fokker-Planck equation [2], [3].