Regularity for solutions to inhomogeneous degenerate parabolic p-Laplace equations
Abstract
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 W1,∞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.
Related
- Local regularity for solutions to singular and degenerate quasilinear parabolic equations
- Asymptotic gradient regularity for degenerate or singular fully nonlinear parabolic equations
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
- Local behavior of solutions to inhomogeneous singular parabolic p-Laplace equations
- Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation
BibTeX
@misc{hao2026regularity,
title = {{Regularity for solutions to inhomogeneous degenerate parabolic $p$-Laplace equations}},
author = {Xia Hao and Yan Li and Zhiwen Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01724},
eprint = {2610.01724},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01724},
}
RIS
TY - UNPB
TI - Regularity for solutions to inhomogeneous degenerate parabolic p-Laplace equations
AU - Xia Hao
AU - Yan Li
AU - Zhiwen Zhao
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01724
UR - https://arxiv.org/abs/2610.01724
AB - 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,∞_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.
ER -