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Peer-reviewedCC BY 4.0Journal of Evolution Equations

Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds

  • Khalid Baadi
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Abstract

In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial A2-weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser’s L2-L∞ estimates for local weak solutions. In the special case of real coefficients, Moser's L2-L∞ estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.

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Cite

BibTeXRIS
BibTeX
@article{baadi2026degenerate,
  title = {{Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds}},
  author = {Khalid Baadi},
  journal = {Journal of Evolution Equations},
  year = {2026},
  month = oct,
  doi = {10.1007/s00028-026-01201-1},
  eprint = {2503.07569},
  archivePrefix = {arXiv},
  url = {https://cnrs.hal.science/hal-04985360v4},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Degenerate parabolic equations in divergence form: fundamental solution and Gaussian bounds
AU  - Khalid Baadi
PY  - 2026
DA  - 2026/10/07
JO  - Journal of Evolution Equations
DO  - 10.1007/s00028-026-01201-1
UR  - https://cnrs.hal.science/hal-04985360v4
AB  - In this paper, we consider second order degenerate parabolic equations with complex, measurable, and time-dependent coefficients. The degenerate ellipticity is dictated by a spatial A_2-weight. We prove that having a generalized fundamental solution with upper Gaussian bounds is equivalent to Moser’s L^2-L^∞ estimates for local weak solutions. In the special case of real coefficients, Moser's L^2-L^∞ estimates are known, which provide an easier proof of Gaussian upper bounds, and a known Harnack inequality is then used to derive Gaussian lower bounds.
SN  - 1424-3199
SN  - 1424-3202
ER  -