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arXiv · 2604.13669

Sharp asymptotic behaviour of symmetric and non-symmetric solutions of the Heat Equation in the Hyperbolic Space

José Alfredo Cañizo·Alejandro Gárriz·Diego Alfonso Marín
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Abstract

In this work we study the large-time behaviour of solutions of the Heat Equation in the hyperbolic space $\mathbb{H}^d$, providing precise speeds of convergence in $L^1$ and $L^\infty$ to their asymptotic profiles by means of an adaptation of entropy estimates. For $L^1$ initial conditions we are able to identify the asymptotic profile in $L^1$, which is not universal but contains a certain memory of the initial distribution of the mass of the solution. We improve thus on previous results, where speed of convergence was absent and asymptotic profiles where not known in the general case, and show a way to adapt entropy estimates employed in the study of diffusion processes to non-compact Riemannian manifolds. The main strategy to prove this is to consider transient profiles as minimizers of the entropy functional. These profiles are time-dependent and encompass the geometric information of the Riemannian manifold.

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José Alfredo Cañizo, Alejandro Gárriz, Diego Alfonso Marín. 2026-04-15. Sharp asymptotic behaviour of symmetric and non-symmetric solutions of the Heat Equation in the Hyperbolic Space. https://arxiv.org/abs/2604.13669

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