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

$$\begin{aligned} \partial _tu-{{\,\textrm{div}\,}}_x \nabla _\xi f(\nabla u)=0 \end{aligned}$$

in a space-time cylinder \(\Omega _T=\Omega \times (0,T)\), subject to time-dependent boundary data \(g:\partial _{\mathcal {P}}\Omega _T\rightarrow \mathbb {R}\) 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 \(g(\cdot ,t)\) 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.