An evolutionary Haar-Rado type theorem

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