About smoothness of solutions of the heat equations in closed,smooth space‐time domains

We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ?² = {(x¹, x²)} of class C^2k. We give sufficient conditions for u to have k^th‐order continuous derivatives with respect to (x¹, x²) in D? for integers k ≥ 2. The equation is supplemented with C^2k boundary data, and we assume that f ? C^2(k?1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc.