Asymptotic L1-decay of Solutions of the Porous Medium Equation to Self-similarity

We consider the flow of gas in an N-dimensional porous medium with initial density v0(x) ≥ 0. The density v(x, t) then satisfies the nonlinear degenerate parabolic equation vt = ∆vm where m > 1 is a physical constant. Assuming that ∫ (1 + |x|2)v0(x)dx < ∞, we prove that v(x, t) behaves asymptotically, as t → ∞, like the Barenblatt-Pattle solution V(|x|, t). We prove that the L1-distance decays at a rate t1/((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].