On the heat diffusion starting with degeneracy

Abstract

In this article we study the instant smoothing property of the heat diffusion that starts with degeneracy: u(t)(t, x) = t(alpha) Delta u + f(t, x), t is an element of (0, T), x is an element of R-d ; u(0, x) = u(0)(x), where alpha is an element of (-1, infinity). We provide the existence and uniqueness result in an appropriate Sobolev space setting. For a fixed f the regularity improvement in Sobolev regularity from u(0) to u changes continuously along a. In particular, the larger alpha > 0, the smaller the improvement is. Moreover, we study a regularity relation between f and u near time t = 0 as alpha varies. (C) 2016 Elsevier Inc. All rights reserved.