Rigidity for the heat equation with density on Riemannian manifolds through a conformal change
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913946796032000 |
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| author | Grigor'yan, Alexander Meglioli, Giulia Roncoroni, Alberto |
| author_facet | Grigor'yan, Alexander Meglioli, Giulia Roncoroni, Alberto |
| contents | We investigate uniqueness of solution to the heat equation with a density $ρ$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_ϕ$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $ρ$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $ρ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_13230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rigidity for the heat equation with density on Riemannian manifolds through a conformal change Grigor'yan, Alexander Meglioli, Giulia Roncoroni, Alberto Analysis of PDEs We investigate uniqueness of solution to the heat equation with a density $ρ$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_ϕ$. We distinguish between the cases $p > 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $ρ$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $ρ$. |
| title | Rigidity for the heat equation with density on Riemannian manifolds through a conformal change |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.13230 |