Singular limit of solutions of the very fast diffusion equation

We prove that the distribution solutions of the very fast diffusion equation partial derivative(u)/partial derivative(t) = Delta(u(m)/m), u > 0, in R-n x (0, infinity), u(x, 0) = u(0)(x) in R-n, where m < 0, n >= 2, constructed in [P. Daskalopoulos, M.A. Del Pino, On nonlinear parabolic equations of very fast diffusion, Arch. Ration. Mech. Anal. 137 (1997) 363-380] are actually classical maximal solutions of the problem. Under the additional assumption that u(0) L-1 (R-n), 0 <= u(0) is an element of L-loc(p) (R-n) for some constant p > n/2, and u(0)(x) > epsilon/vertical bar x vertical bar(2 alpha) for any vertical bar x vertical bar > R-1 where epsilon > 0, R-1 > 0, m(0) < 0, alpha < min(l/(l - m(0)), 1/vertical bar m(0)vertical bar), are constants satisfying p > (1 - m(0))n/2, we prove that the solution of the above problem will converge uniformly on every compact subset of R-n x (0, infinity) to the maximal solution of the equation v(t) = Delta log v, v(x, 0) = u(0)(x), as m NE arrow 0(-). For any smooth bounded domain Omega C R-n, m(0) < 0, m is an element of [m(0), 0) boolean OR (0, 1), and 0 <= u(0) is an element of L-p (Omega) for some constant p > (1 - m(0)) max(l, n/2), we prove the existence and uniqueness of solutions of the Dirichlet problem partial derivative u/partial derivative t = Delta(u(m)/m), u > 0, in Omega x (0, infinity), u = u(0) in Omega, u = g on partial derivative Omega x (0, infinity) with either finite or infinite positive boundary value g. We also prove a similar convergence result for the solutions of the above Dirichlet problem as m -> 0. (C) 2006 Elsevier Ltd. All rights reserved.