An evolutionary Haar-Rado type theorem

Abstract

In this paper, we study variational solutions to parabolic equations of the type ∂t u −divx (Dξ f (Du))+ Du g(x, u) = 0, where u attains time-independent boundary values u0 on the parabolic boundary and f, g fulfill convexity assumptions. We establish a Haar-Rado type theorem: If the boundary values u0 admit a modulus of continuity ω and the estimate |u(x, t)−u0(γ )| ≤ ω(|x −γ |) holds, then u admits the same modulus of continuity in the spatial variable.