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 $$\begin{aligned} \partial _tu-{{\,\textrm{div}\,}}_x \nabla _\xi f(\nabla u)=0 \end{aligned}$$ ∂ t u - div x ∇ ξ f ( ∇ u ) = 0 in a space-time cylinder $$\Omega _T=\Omega \times (0,T)$$ Ω T = Ω × ( 0 , T ) , subject to time-dependent boundary data $$g:\partial _{\mathcal {P}}\Omega _T\rightarrow \mathbb {R}$$ g : ∂ P Ω T → 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 $$\partial \Omega \times (0,T)$$ ∂ Ω × ( 0 , T ) . More precisely, we require that, for each fixed $$t\in [0,T)$$ t ∈ [ 0 , T ) , the graph of $$g(\cdot ,t)$$ g ( · , t ) over $$\partial \Omega $$ ∂ Ω 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.