A Priori estimates for very fast diffusion equations in R-n

A priori estimates are established for the very fast diffusion equation -pa rtial derivative u/partial derivative t + Delta(u(m)/m) = 0 in R-n x (0, T) where m < 0 in the form of necessary conditions on the initial value u(0)( x) and the time level T. The demonstration is based on the volumetric mean of the solutions. Also, given are a priori estimates for the related ellipt ic equation lambda mu Delta upsilon + upsilon(-mu) = f(x) in R-n where mu > 0 and lambda > 0 in the form of necessary conditions on the non-homogeneou s term f(x) and the parameter lambda. Necessary conditions on the domain of the evolution operator follows from these results. Copyright (C) 2000 John Wiley & Sons, Ltd.