Asymptotic L^1-decay of solutions of the porous medium equation to self-similarity
José Antonio Carrillo, Giuseppe Toscani · Indiana University Mathematics Journal · 2000
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].