Asymptotic behaviour for the heat equation in hyperbolic space
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Publication:6136047
DOI10.4310/CAG.2022.V30.N9.A7zbMATH Open1528.35219arXiv1811.09034OpenAlexW2901719634MaRDI QIDQ6136047
Publication date: 28 August 2023
Published in: Communications in Analysis and Geometry (Search for Journal in Brave)
Abstract: Following the classical result of long-time asymptotic convergence towards the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space , we examine the question of long-time behaviour of the Heat Equation in the Hyperbolic Space , , also for integrable solutions. We show that the typical convergence proof towards the fundamental solution works in the class of radially symmetric solutions. We also prove the more precise result that says that this limit behaviour is exactly described by the 1D Euclidean kernel, but only after correction of a remarkable outward drift with constant speed produced by the geometry. Finally, we find that such fine convergence results are false for general nonnegative solutions with integrable initial data.
Full work available at URL: https://arxiv.org/abs/1811.09034
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