On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915651817308160 |
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| author | Baadi, Khalid |
| author_facet | Baadi, Khalid |
| contents | 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 $L^2$-$L^\infty$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03978 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms Baadi, Khalid Analysis of PDEs Functional Analysis Primary: 35K65, 35K10, 35A08, 35K15 Secondary: 35K40, 26A33 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 $L^2$-$L^\infty$ 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. |
| title | On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms |
| topic | Analysis of PDEs Functional Analysis Primary: 35K65, 35K10, 35A08, 35K15 Secondary: 35K40, 26A33 |
| url | https://arxiv.org/abs/2512.03978 |