An evolutionary Haar-Rado type theorem

Abstract

In this paper, we study variational solutions to parabolic equations of the type ∂tu- div x(Dξf(Du)) + Dug(x, u) = 0 , where u attains time-independent boundary values u on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values u admit a modulus of continuity ω and the estimate | u(x, t) - u(γ) | ≤ ω(| x- γ|) holds, then u admits the same modulus of continuity in the spatial variable.