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Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

Abstract : This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-$K$-invariant $L^{1}$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-$K$-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-$K$-invariance condition and strong $L^{\infty}$ convergence.
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https://hal.archives-ouvertes.fr/hal-03459571
Contributor : Hong-Wei Zhang Connect in order to contact the contributor
Submitted on : Wednesday, December 1, 2021 - 1:52:22 AM
Last modification on : Tuesday, January 11, 2022 - 5:56:35 PM

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  • HAL Id : hal-03459571, version 1

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Jean-Philippe Anker, Effie Papageorgiou, Hong-Wei Zhang. Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces. 2021. ⟨hal-03459571⟩

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