Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

We establish the existence of Lipschitz-continuous solutions to the Cauchy-Dirichlet problem for a class of evolutionary partial differential equations of the form $\begin{array}{c}{\partial }_{t}u-{\text{div}}_x{\nabla }_{\xi }f\left(\nabla u\right)=0\end{array}$ in a space-time cylinder ${\Omega }_T=\Omega \times (0,T)$ , subject to time-dependent boundary data $g:{\partial }_P{\Omega }_T\to R$ prescribed on the parabolic boundary. The main novelty in our analysis is a time-dependent version of the classical bounded slope condition, imposed on the boundary data g along the lateral boundary $\partial \Omega \times (0,T)$ . More precisely, we require that, for each fixed $t\in [0,T)$ , the graph of $g(·,t)$ over $\partial \Omega$ admits supporting hyperplanes with slopes that may vary in time but remain uniformly bounded. The key to handling time-dependent data lies in constructing more flexible upper and lower barriers.