LOCAL SPACE - ANALITICITY OF SOLUTIONS OF CERTAIN SINGULAR PARABOLIC EQUATIONS

The equation in (1.1) below, is singular at points where u = 0. We investigate the behaviour of the solution near these points of singularity, when m is in the range (1.2). It is shown that in spite of the singularity of the p.d.e., non-negative solutions are analytic in the space variables and at least Lipschitz continuous in t. We also establish sharp decay rates near the boundary of their domain of definition and near the extinction time. These results follow from accurate upper and lower bounds on the solutions that can be regarded as some sort of a global Harnack principle. The range in (1.2) is the best possible for such a Harnack principle to hold.