Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

Abstract

We establish the existence of Lipschitz continuous solutions to the Cauchy Dirichlet problem for a class of evolutionary partial differential equations of the form $${\partial }_{t}u-{\text{div}}_x{\nabla }_{\hat{I}}¾f(\nabla u)=0$$ in a space-time cylinder $\hat{I}{\text{c◯}}_T=\hat{I}\text{c◯}\times (0,T)$, subject to time-dependent boundary data $g:{\partial }_{\mathcal{P}}\hat{I}{\text{c◯}}_T\to \mathbf{R}$ 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 $\partial \hat{I}\text{c◯}\times (0,T)$. More precisely, we require that for each fixed $t\in [0,T)$, the graph of $g(\cdot ,t)$ over $\partial \hat{I}\text{c◯}$ 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.

37 pages, 2 figures