We establish the existence of Lipschitz continuous solutions to the Cauchy
Dirichlet problem for a class of evolutionary partial differential equations of
the form $$ \partial_tu-\text{div}_x \nabla_\xi f(\nabla u)=0 $$ in a
space-time cylinder $\Omega_T=\Omega\times (0,T)$, subject to time-dependent
boundary data $g\colon \partial_{\mathcal{P}}\Omega_T\to \mathbf{R}$ 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 $\partial\Omega\times (0,T)$. More precisely, Noi
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.

Questo articolo esplora i giri e le loro implicazioni.

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