标题： 具有动态数据的抛物型偏微分方程在有界斜率条件下的研究 显示英文标题

标题： Parabolic PDEs with Dynamic Data under a Bounded Slope Condition

摘要： 我们建立了Lipschitz连续解的存在性，这些解满足一类形式为$$ \partial_tu-\text{div}_x \nabla_\xi f(\nabla u)=0 $$的发展型偏微分方程的柯西-狄利克雷问题，定义在一个时空柱状区域$\Omega_T=\Omega\times (0,T)$中，并且以时间相关的边界数据$g\colon \partial_{\mathcal{P}}\Omega_T\to \mathbf{R}$作为抛物边界上的约束条件。分析中的主要新颖之处在于对边界数据$g$在横向边界$\partial\Omega\times (0,T)$上施加了一个时间相关版本的经典有界斜率条件。 更精确地说，我们要求对于每个固定的$t\in [0,T)$，$g(\cdot ,t)$在$\partial\Omega$上的图像允许支撑超平面，其斜率可以随时间变化但始终保持一致有界。 处理时变数据的关键在于构建更灵活的上下屏障。 显示英文摘要

摘要： 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, 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.