On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
Abstract
In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded lower-order terms. The unbounded lower-order terms are allowed to lie in mixed time-space Lebesgue or even Lorentz spaces. Our notion of weak solutions is formulated under minimal assumptions. We prove the existence and uniqueness of a fundamental solution, which coincides with the associated evolution family for the homogeneous problem (i.e., with zero source term) and provides a representation formula for all weak solutions. We also establish L2 off-diagonal estimates for the fundamental solution and derive Gaussian upper bounds under the weak assumption of Moser's L2-L∞ estimates for weak solutions. Our approach is purely variational and avoids any a priori regularity assumptions on weak solutions or regularization via smooth approximations. Two key ingredients are norm inequalities for fractional powers of the degenerate Laplacian, and a set of embeddings that ensure time continuity of weak solutions, extending the classical Lions regularity theorem and accommodating a wide class of source terms.
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BibTeX
@article{baadi2026wellposedness,
title = {{On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms}},
author = {Khalid Baadi},
journal = {Advances in Calculus of Variation},
year = {2026},
month = sep,
eprint = {2512.03978},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05399830v2},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
AU - Khalid Baadi
PY - 2026
DA - 2026/09/29
JO - Advances in Calculus of Variation
UR - https://hal.science/hal-05399830v2
AB - In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded lower-order terms. The unbounded lower-order terms are allowed to lie in mixed time-space Lebesgue or even Lorentz spaces. Our notion of weak solutions is formulated under minimal assumptions. We prove the existence and uniqueness of a fundamental solution, which coincides with the associated evolution family for the homogeneous problem (i.e., with zero source term) and provides a representation formula for all weak solutions. We also establish L^2 off-diagonal estimates for the fundamental solution and derive Gaussian upper bounds under the weak assumption of Moser's L^2-L^∞ estimates for weak solutions. Our approach is purely variational and avoids any a priori regularity assumptions on weak solutions or regularization via smooth approximations. Two key ingredients are norm inequalities for fractional powers of the degenerate Laplacian, and a set of embeddings that ensure time continuity of weak solutions, extending the classical Lions regularity theorem and accommodating a wide class of source terms.
SN - 1864-8266
SN - 1864-8266
ER -