Asymptotic behaviour for the heat equation in hyperbolic space

Following the classical result of long-time asymptotic convergence towards a multiple of the Gaussian kernel that holds true for integrable solutions of the Heat Equation posed in the Euclidean Space $\mathbb{R}^n$, we examine the question of long-time behaviour of the Heat Equation in the Hyperbolic Space $\mathbb{H}^n, n \gt 1$, also for integrable data and solutions. We show that the typical convergence proof towards a multiple of 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 simple 1D Euclidean kernel after a fortunate change of variables. Indeed, this counter-intuitive fact happens after introducing the strong correction caused by a remarkable outward drift with constant speed (ballistic behaviour), an effect produced by the geometry. Finally, we find that such fine convergence results are false for general nonnegative solutions with integrable initial data if the radial symmetry is missing.

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The author’s was partially funded by Project MTM2014-52240-P (Spain). Partially performed as an Honorary Professor at Univ. Complutense de Madrid.

Received 17 March 2019

Accepted 7 April 2020

Published 17 August 2023