Holder estimates for singular non-local parabolic equations

In this paper, we establish local Holder estimate for non-negative solutions of the singular equation (M.P) below, for in in the range of exponents (n-2 sigma/n+2 sigma, 1). Since we have trouble in finding the local energy inequality of upsilon directly, we use the fact that the operator (-Delta)(sigma) can be thought as the normal derivative of some extension upsilon* of upsilon to the upper half space (Caffarelli and Silvestre, 2007 [5]), i.e., upsilon is regarded as boundary value of upsilon* the solution of some local extension problem. Therefore, the local Holder estimate of upsilon can be obtained by the same regularity of upsilon*. In addition, it enables us to describe the behavior of solution of non-local fast diffusion equation near their extinction time. (C) 2011 Elsevier Inc. All rights reserved.