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

TL;DR

This paper analyzes the long-term behavior of solutions to the Heat Equation in hyperbolic space, providing precise convergence speeds and identifying non-universal asymptotic profiles influenced by initial conditions.

Contribution

It advances understanding by establishing convergence rates and explicit asymptotic profiles for solutions in hyperbolic space, using adapted entropy methods.

Findings

Identified non-universal asymptotic profiles containing initial distribution memory.

Established convergence speeds in both L^1 and L^∞ norms.

Extended entropy estimate techniques to non-compact Riemannian manifolds.

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.