| Abstract: |
| We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with time-dependent boundary data and establish the existence of Lipschitz continuous solutions under minimal assumptions.
The main novelty is the introduction of a genuinely time-dependent variant of the classical bounded slope condition. In contrast to existing approaches, this condition allows the supporting hyperplanes of the boundary data to vary in time while remaining uniformly bounded. This yields a flexible geometric framework that accommodates non-stationary boundary values and extends previous results beyond the time-independent setting.
Our proof is based on a new barrier construction adapted to the parabolic geometry and the time-dependent boundary behavior. This approach provides control up to the boundary and leads to global Lipschitz bounds for solutions. The method is robust and may be applicable to related parabolic problems.
This work was carried out in collaboration with Giulia Treu (University of Padova) and Verena Boegelein (University of Salzburg). |
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