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 $${\mathrm{\partial }}_{t}u-{\text{div}}_x{\mathrm{\nabla }}_{\xi }f(\mathrm{\nabla }u)=0$$ in a space-time cylinder ${\mathrm{\Omega }}_T=\mathrm{\Omega }\times (0,T)$, subject to time-dependent boundary data $g\text{\hspace{0.17em}⁣}:{\mathrm{\partial }}_{\mathcal{P}}{\mathrm{\Omega }}_T\to \mathbf{R}$ 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 $\mathrm{\partial }\mathrm{\Omega }\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\mathrm{\partial }\mathrm{\Omega }$ 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.