A central limit theorem for solutions of the porous medium equation

We study the large-time behavior of the second moment (energy) E(t) = 1/2 integral vertical bar x vertical bar(2)upsilon(x, t)dx for the flow of a gas in a N-dimensional porous medium with initial density upsilon(0)(x) >= 0. The 2 density upsilon(x, t) satisfies the nonlinear degenerate parabolic equation upsilon(t) = Delta upsilon(m) where m > 1 is a physical constant. Assuming that integral(upsilon(0)(m) (x) + vertical bar x vertical bar(2+delta) upsilon(0)(x))dx < infinity for some delta > 0, we prove that E(t) behaves asymptotically, as t -> infinity, like the energy E-B(t) of the Barenblatt-Pattle solution B(vertical bar x vertical bar, t). This is shown by proving that E(t)/E-B(t) converges to 1 at the (optimal) rate t(-2/(N(m-)+2)). A simple corollary of this result is a central limit theorem for the scaled solution E(t)(N/2)upsilon(E (t)(1/2)x, t).