发表机构
Hebei Normal University; Beijing Normal University; University of Science and Technology Beijing（河北师范大学; 北京师范大学; 北京科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对非齐次退化 $p$-Laplace 方程，通过建立时间 Lipschitz 估计直接推导水平集测度向前传播，绕开能量不等式，证明 Hölder 正则性，并给出时空 $W^{1,\infty}$ 估计及反例说明正等待时间条件的尖锐性。
AI 中文摘要
对于 $p$-Laplace 型拟线性退化抛物方程的弱解，应用内在尺度方法证明其 Hölder 正则性的一个核心障碍是推导水平集空间测度的向前时间传播估计。本文针对具有有界非负时间无关强迫项和初始数据的非齐次退化 $p$-Laplace 方程，重新审视并克服了这一困难。我们的新证明基于在任意与初始时间间隔正时间的区间上建立的时间 Lipschitz 估计。该估计提供的强正则性增益使我们能够直接推导水平集测度的向前传播，从而绕过了先前方法中作为起点的能量不等式。此外，我们建立了远离初始时间的时空 $W^{1,\infty}_{x,t}$ 估计，并构造反例以证明在仅有界初始数据下正等待时间条件的尖锐性。
英文摘要
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their Hölder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcome this difficulty for an inhomogeneous degenerate parabolic $p$-Laplace equation with bounded nonnegative time-independent forcing term and initial data. Our new proof is built upon the temporal Lipschitz estimate that we establish on any time interval separated from the initial time by a positive amount. The strong regularity gain furnished by this estimate allows us to derive the forward propagation of level-set measures directly, thereby bypassing the energy inequality that served as the starting point of the previous approach. Furthermore, we establish space-time $W^{1,\infty}_{x,t}$ estimates away from the initial time and construct counterexamples to demonstrate the sharpness of the positive waiting-time condition under merely bounded initial data.
