Asymptotics of the fast-diffusion equation with critical exponent

We study the large-time behavior of the solutions of the initial-value prob lem for the nonlinear diffusion equation (ND) u(t) =del . (u(-sigma)del u) in R-n x Rin dimensions n greater than or equal to 3 with nonnegative initial data u (x, 0) is an element of L-1 (R-n) when the exponent takes on the critical v alue sigma = 2/n. This represents a borderline case in the study of the pro blem and offers marked qualitative and technical differences with the neigh boring cases sigma approximate to 2/n, sigma not equal 2/n. In particular, it marks the transition between two completely different asymptotic behavio r types. It is known that solutions exist globally in time and conserve the L-1-norm for this problem. We prove that they decay exponentially in time with a complicated law: log parallel to u (., t)parallel to(infinity)similar to -kM(-2/(n-2)) t(n/( n-2)) as t-->infinity, where M = integral u (x, 0)dx is the conserved mass and the constant k > 0 depends only on the dimension n. This strongly differs from the comparative ly simple self-similar asymptotics of the case sigma < 2/n. The description is split into an inner and an outer region, conveniently ma tched at a transition layer. The analysis of the outer region can be done i ndependently and the behavior is governed by a first-order conservation law which acts as the reduced asymptotic equation. The uniqueness theory for f irst-order conservation laws is one of the great contributions of S. N. Kru zhkov to mathematics. The behavior in the inner parabolic region is then st udied by means of a semiconvexity argument which makes it possible to trans late into this region the precise behavior from the outer region.