Asymptotic Behavior of the Heat Semigroup on Certain Riemannian Manifolds

Grigor’yan, Alexander; Papageorgiou, Effie; Zhang, Hong-Wei

doi:10.1007/978-3-031-37800-3_8

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…The connection between the long-time behavior of the solution u(t, x) of (1.1) for initial data u 0 ∈ L 1 (M, μ) (where μ is the Riemannian measure on M) and that of the heat kernel h t (x, y) has recently been the subject of extensive studies, see, for example, [7,19,30] or see [1,2,7,23] for variants and related questions. Denote by M = M dμ(x) u 0 (x) the mass of the initial data.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Papageorgiou

2024

J. Evol. Equ.

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This work deals with the extension problem for the fractional Laplacian on Riemannian symmetric spaces G/K of noncompact type and of general rank, which gives rise to a family of convolution operators, including the Poisson operator. More precisely, motivated by Euclidean results for the Poisson semigroup, we study the long-time asymptotic behavior of solutions to the extension problem for $$L^1$$ L 1 initial data. In the case of the Laplace–Beltrami operator, we show that if the initial data are bi-K-invariant, then the solution to the extension problem behaves asymptotically as the mass times the fundamental solution, but this convergence may break down in the non-bi-K-invariant case. In the second part, we investigate the long-time asymptotic behavior of the extension problem associated with the so-called distinguished Laplacian on G/K. In this case, we observe phenomena which are similar to the Euclidean setting for the Poisson semigroup, such as $$L^1$$ L 1 asymptotic convergence without the assumption of bi-K-invariance.

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Section: Introductionmentioning

confidence: 99%

“…Similar results were obtained in [7] in a more general setting of symmetric spaces of noncompact type by using tools of harmonic analysis. In [19], it was shown that (1.3) fails on connected sums R n #R n , n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Papageorgiou

2024

J. Evol. Equ.

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Asymptotic Behavior of Solutions to the Extension Problem for the Fractional Laplacian on Noncompact Symmetric Spaces

Papageorgiou

2024

Trends in Mathematics

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Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

Papageorgiou

2023

Potential Anal

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This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $$L^1$$ L 1 initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

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Asymptotic Behavior of the Heat Semigroup on Certain Riemannian Manifolds

Cited by 4 publications

References 17 publications

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Asymptotic behavior of solutions to the extension problem for the fractional Laplacian on noncompact symmetric spaces

Asymptotic Behavior of Solutions to the Extension Problem for the Fractional Laplacian on Noncompact Symmetric Spaces

Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds

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