PreprintCC BY-NC 4.0HAL
SINGULAR SOLUTIONS TO PARABOLIC EQUATIONS IN ONE SPACE DIMENSION
Abstract
We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every p>6, there exists such an equation admitting an energy solution that fails to be p-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi--Nash--Moser theory fails for complex coefficients already in one space dimension.
Related
- Local behavior of solutions to inhomogeneous singular parabolic p-Laplace equations
- 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
- Stochastic Itô Equations and Parabolic Second-Order Equations with singular Drift
BibTeX
@misc{auscher2026singular,
title = {{SINGULAR SOLUTIONS TO PARABOLIC EQUATIONS IN ONE SPACE DIMENSION}},
author = {Pascal Auscher and Moritz Egert and Benjamin W. Kosmala},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17351},
eprint = {2609.17351},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05739311v1},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - SINGULAR SOLUTIONS TO PARABOLIC EQUATIONS IN ONE SPACE DIMENSION
AU - Pascal Auscher
AU - Moritz Egert
AU - Benjamin W. Kosmala
PY - 2026
DA - 2026/09/04
PB - HAL
DO - 10.48550/arxiv.2609.17351
UR - https://hal.science/hal-05739311v1
AB - We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every p>6, there exists such an equation admitting an energy solution that fails to be p-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi--Nash--Moser theory fails for complex coefficients already in one space dimension.
ER -