Local regularity for solutions to singular and degenerate quasilinear parabolic equations
Abstract
The main purpose of this paper is to prove local Hölder continuity for weak solutions to quasilinear parabolic equations of p-Laplace type under finite integrability assumptions on the forcing, covering the supercritical singular range and the entire degenerate regime. The key ingredient is an L∞-valued temporal Hölder estimate of the solution after any fixed positive waiting time, which is achieved by coupling a time-rescaling argument with De Giorgi iteration. We also provide counterexamples involving forcing terms with weaker spatial integrability to show that even bounded continuous solutions with compatible data may possess no positive temporal Hölder exponent. In particular, these examples establish the sharpness of the spatial integrability threshold in the degenerate case. Finally, a separate counterexample is constructed to demonstrate that the temporal Lipschitz endpoint can fail for bounded continuous forcing when it depends on time.
Related
- Regularity for solutions to inhomogeneous degenerate parabolic p-Laplace 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
- Solutions to fourth order degenerate parabolic equations obtained via weighted energy dissipation
- Hölder regularity of solutions to parabolic equations associated with fractional p-Laplacian type operators
BibTeX
@misc{li2026local,
title = {{Local regularity for solutions to singular and degenerate quasilinear parabolic equations}},
author = {Yan Li and Zhiwen Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37538},
eprint = {2610.37538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37538},
}
RIS
TY - UNPB
TI - Local regularity for solutions to singular and degenerate quasilinear parabolic equations
AU - Yan Li
AU - Zhiwen Zhao
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37538
UR - https://arxiv.org/abs/2610.37538
AB - The main purpose of this paper is to prove local Hölder continuity for weak solutions to quasilinear parabolic equations of p-Laplace type under finite integrability assumptions on the forcing, covering the supercritical singular range and the entire degenerate regime. The key ingredient is an L^∞-valued temporal Hölder estimate of the solution after any fixed positive waiting time, which is achieved by coupling a time-rescaling argument with De Giorgi iteration. We also provide counterexamples involving forcing terms with weaker spatial integrability to show that even bounded continuous solutions with compatible data may possess no positive temporal Hölder exponent. In particular, these examples establish the sharpness of the spatial integrability threshold in the degenerate case. Finally, a separate counterexample is constructed to demonstrate that the temporal Lipschitz endpoint can fail for bounded continuous forcing when it depends on time.
ER -