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

Alejandro G\'arriz; Diego Alfonso Mar\'in; Jos\'e Alfredo Ca\~nizo

arxiv: 2604.13669 · v1 · submitted 2026-04-15 · 🧮 math.AP

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

Jos\'e Alfredo Ca\~nizo , Alejandro G\'arriz , Diego Alfonso Mar\'in This is my paper

Pith reviewed 2026-05-10 13:03 UTC · model grok-4.3

classification 🧮 math.AP

keywords heat equationhyperbolic spaceasymptotic behaviourentropy estimateslarge-time behaviourL1 convergencenon-compact manifolds

0 comments

The pith

Solutions of the heat equation in hyperbolic space converge at sharp rates to non-universal profiles that remember the initial mass distribution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates the large-time behaviour of solutions to the heat equation posed on hyperbolic space. It proves that for initial conditions with finite total mass the asymptotic profile in the L1 norm is determined by the initial distribution of that mass rather than being independent of the starting point. Explicit rates of convergence are given in both the L1 and L infinity norms through an adaptation of entropy estimates. The central technique is to view time-dependent profiles as minimizers of an entropy functional that takes into account the geometry of the manifold. This adaptation shows how such methods can be carried over to non-compact Riemannian manifolds.

Core claim

The paper shows that solutions to the heat equation on the hyperbolic space H^d approach time-dependent transient profiles that minimize an entropy functional. These profiles carry a memory of the initial mass distribution for general L1 data and yield sharp rates of convergence in L1 and L∞, improving on earlier work that lacked rates and general profiles.

What carries the argument

Time-dependent transient profiles considered as minimizers of the entropy functional on the hyperbolic manifold.

Load-bearing premise

That the transient profiles function as entropy minimizers on the non-compact hyperbolic manifold to deliver the claimed sharp convergence rates.

What would settle it

A specific initial datum in L1 for which the L1 distance to the predicted asymptotic profile decays slower than the rate given in the paper.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Sharp rates and non-universal L1 profiles for the heat equation on hyperbolic space via adapted entropy estimates, but the coercivity of the functional on the non-compact manifold needs explicit verification.

read the letter

The main things to know are that this paper establishes sharp rates of convergence in both L1 and L^infty for solutions of the heat equation on hyperbolic space, and it identifies the asymptotic profile in L1 as non-universal, carrying some memory of how the initial mass was distributed. They arrive at this by adapting entropy estimates, treating the time-dependent transient profiles as minimizers of the entropy functional that accounts for the manifold geometry. This improves on earlier work by supplying the missing speeds and the profile details for general L1 data. The paper does well in laying out a strategy to move entropy methods over to non-compact Riemannian manifolds without losing the geometric content. The soft spot worth noting is around the entropy minimization itself. On hyperbolic space the volume grows exponentially, so the entropy integral may not have minimizers or stable ones for every L1 initial condition without further restrictions on decay or moments. If the paper does not show that the second variation controls the behavior at infinity uniformly in time, then the sharpness of the rates and the memory effect could be less robust than claimed. The stress-test concern lands here and needs checking against the actual proofs. Assuming the derivations handle those tails properly, the rest of the argument looks standard and consistent. This work is aimed at researchers in geometric PDEs and asymptotic analysis on manifolds. Someone already familiar with entropy techniques for diffusion equations would get the most out of the adaptation and the specific rates. It deserves a serious referee. The improvements are concrete enough and the setting is interesting, so it should receive full review. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims to study the large-time behaviour of solutions of the Heat Equation in the hyperbolic space H^d, providing precise speeds of convergence in L1 and L^infty to their asymptotic profiles by means of an adaptation of entropy estimates. For L1 initial conditions, the asymptotic profile in L1 is identified as non-universal, containing a memory of the initial distribution of the mass. The main strategy 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.

Significance. If the adaptation of entropy estimates is rigorously justified, the results would improve on prior work by supplying both convergence rates and explicit (non-universal) asymptotic profiles for general L1 data on a non-compact manifold. The approach of treating time-dependent transient profiles as entropy minimizers could serve as a template for diffusion problems on other Riemannian manifolds with exponential volume growth.

major comments (2)

[Main strategy / entropy-minimization argument] The central strategy of treating time-dependent transient profiles as minimizers of the adapted entropy functional (described in the abstract and main strategy paragraph) requires a self-contained proof of existence, uniqueness, and stability of these minimizers on the non-compact H^d. The exponential volume growth prevents standard coercivity, so uniform tail control must be established for arbitrary L1 data; without it the claimed sharp L1/L^infty rates and the memory effect in the profile rest on an unverified assumption.
[Entropy-estimate adaptation and convergence-rate proofs] The adaptation of entropy estimates to obtain precise speeds of convergence must include explicit error bounds or a quantitative second-variation analysis that controls the tails uniformly in time. The abstract states that rates are obtained but supplies no derivation or verification step; this gap is load-bearing for the claim that the rates are sharp and that the profile retains initial-mass memory.

minor comments (2)

[Notation and definitions] Clarify the precise form of the time-dependent entropy functional and how it differs from the Euclidean case; the current description is too high-level for reproducibility.
[Introduction] Add a brief comparison table or paragraph contrasting the new rates and profiles with those in the cited previous results on the heat equation in H^d.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments, which identify key points where additional rigor is needed. We will prepare a major revision that incorporates self-contained proofs and quantitative estimates as requested. Our point-by-point responses follow.

read point-by-point responses

Referee: [Main strategy / entropy-minimization argument] The central strategy of treating time-dependent transient profiles as minimizers of the adapted entropy functional (described in the abstract and main strategy paragraph) requires a self-contained proof of existence, uniqueness, and stability of these minimizers on the non-compact H^d. The exponential volume growth prevents standard coercivity, so uniform tail control must be established for arbitrary L1 data; without it the claimed sharp L1/L^infty rates and the memory effect in the profile rest on an unverified assumption.

Authors: We acknowledge that the manuscript's treatment of the time-dependent entropy minimizers, while motivated by the geometry of H^d, does not yet contain a fully self-contained proof of existence, uniqueness, and stability together with uniform tail control for general L^1 data. The exponential volume growth indeed precludes direct application of standard coercivity arguments on Euclidean space. In the revised version we will insert a dedicated subsection that establishes these properties by exploiting the explicit form of the hyperbolic metric and the associated volume measure; the argument will combine a direct minimization procedure with tail-decay estimates derived from the entropy functional itself. This addition will remove the gap and place the sharp rates and the memory effect on a rigorous footing. revision: yes
Referee: [Entropy-estimate adaptation and convergence-rate proofs] The adaptation of entropy estimates to obtain precise speeds of convergence must include explicit error bounds or a quantitative second-variation analysis that controls the tails uniformly in time. The abstract states that rates are obtained but supplies no derivation or verification step; this gap is load-bearing for the claim that the rates are sharp and that the profile retains initial-mass memory.

Authors: We agree that the current presentation of the adapted entropy estimates lacks the explicit quantitative bounds and second-variation analysis needed to verify sharpness and uniform tail control. Although the main body sketches the entropy dissipation identity, the derivation of the precise L^1 and L^∞ rates is not carried out with the required error estimates. In the revision we will add a quantitative second-variation computation around the time-dependent minimizers, together with explicit remainder bounds that are uniform in time; these will be used to obtain the claimed convergence speeds and to confirm that the asymptotic profile retains a memory of the initial mass distribution. revision: yes

Circularity Check

0 steps flagged

No circularity: entropy adaptation relies on external hyperbolic geometry and standard techniques

full rationale

The derivation adapts entropy methods by positing time-dependent transient profiles as minimizers of the entropy functional on H^d, using the manifold's Riemannian volume growth to encode geometry into the profiles. This step is presented as an independent extension of known Euclidean entropy techniques rather than a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations reduce by construction to the inputs; the claimed L1/L^infty rates and non-universal profiles follow from the adaptation without circular reduction. The approach is self-contained against external benchmarks of hyperbolic geometry and entropy dissipation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard properties of the heat equation on Riemannian manifolds and geometry of hyperbolic space; no free parameters, new entities, or ad-hoc assumptions are introduced in the abstract description.

axioms (2)

standard math The heat equation on a Riemannian manifold admits solutions whose large-time behavior can be analyzed via entropy functionals.
Implicit foundation for adapting entropy estimates to H^d.
domain assumption Hyperbolic space H^d has geometric features (negative curvature, volume growth) that allow time-dependent entropy minimizers to capture diffusion asymptotics.
Used to justify the profiles encompassing manifold geometry.

pith-pipeline@v0.9.0 · 5463 in / 1405 out tokens · 37106 ms · 2026-05-10T13:03:28.591606+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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[1] [1]

Panoramas et Synthèses

Ané, C.; Blachère, S.; Chafaï, D.; Fougères, P.; Gentil, I.; Malrieu, F.; Roberto, C., and Scheffer, G.Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses. Société Mathématique de France, Paris, 2000. ISBN 2-85629-105-8

work page 2000

[2] [2]

Heat kernel and green function estimates on noncompact symmetric spaces.Geometric and Functional Analysis, 9:1035–91, 1999

Anker, Jean-Philippe and Ji, Lizhen. Heat kernel and green function estimates on noncompact symmetric spaces.Geometric and Functional Analysis, 9:1035–91, 1999

work page 1999

[3] [3]

Asymptotic be- havior of solutions to the heat equation on noncompact symmetric spaces.Journal of Functional Analysis, 284(6):109828, 2023

Anker, Jean-Philippe; Papageorgiou, Effie, and Zhang, Hong-Wei. Asymptotic be- havior of solutions to the heat equation on noncompact symmetric spaces.Journal of Functional Analysis, 284(6):109828, 2023

work page 2023

[4] [4]

On convex sobolev inequalities and the rate of convergence to equilibrium for fokker-planck type equa- tions.Communications in Partial Differential Equations, 26(1–2):43–100, 2001

Arnold, A.; Markowich, P.; Toscani, G., and Unterreiter, A. On convex sobolev inequalities and the rate of convergence to equilibrium for fokker-planck type equa- tions.Communications in Partial Differential Equations, 26(1–2):43–100, 2001

work page 2001

[5] [5]

Entropies and equilibria of many- particle systems: an essay on recent research.Monatshefte fur Mathematik, 142 (1–2), 2004

Arnold, A.; Carrillo, J.A.; Desvillettes, L.; Dolbeault, J.; Jüngel, A.; Lederman, C.; Markovich, P.A.; Toscani, G., and Villani, C. Entropies and equilibria of many- particle systems: an essay on recent research.Monatshefte fur Mathematik, 142 (1–2), 2004

work page 2004

[6] [6]

Springer, Cham, 2014

Bakry, Dominique; Gentil, Ivan, and Ledoux, Michel.Analysis and geometry of Markov diffusion operators, volume 348 ofGrundlehren der mathematischen Wis- senschaften. Springer, Cham, 2014. ISBN 978-3-319-00226-2; 978-3-319-00227-9

work page 2014

[7] [7]

Bonforte, M.; Dolbeault, J.; Grillo, G., and Vázquez, J. L. Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities.Proc. Natl. Acad. Sci. USA, 107(38):16459–16464, 2010

work page 2010

[8] [8]

AsymptoticL1–decay of solutions of the porous medium equation to self-similarity.Indiana Univ

Carrillo, José Antonio and Toscani, Giuseppe. AsymptoticL1–decay of solutions of the porous medium equation to self-similarity.Indiana Univ. Math. J., 49(1): 113–142, 2000

work page 2000

[9] [9]

Large-time estimates for the dirichlet heat equation in exterior domains.arXiv:2503.01492v1, 2025

Cañizo, José A.; Gárriz, Alejandro, and Quirós, Fernando. Large-time estimates for the dirichlet heat equation in exterior domains.arXiv:2503.01492v1, 2025

work page arXiv 2025

[10] [10]

CambridgeTractsinMathematics

Davies, E.B.Heat Kernels and Spectral Theory. CambridgeTractsinMathematics. Cambridge University Press, 1989

work page 1989

[11] [11]

The lévy-fokker-planck equation:ϕ-entropies and convergence to equilibrium.Asymptotic Analysis, 59(3–4):125–138, 2008

Gentil, Ivan and Imbert, Cyril. The lévy-fokker-planck equation:ϕ-entropies and convergence to equilibrium.Asymptotic Analysis, 59(3–4):125–138, 2008

work page 2008

[12] [12]

American Mathematical Society, Prov- idence, RI; International Press, Boston, MA, 2009

Grigor’yan, Alexander.Heat kernel and analysis on manifolds, volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Prov- idence, RI; International Press, Boston, MA, 2009

work page 2009

[13] [13]

The heat kernel on hyperbolic space.Bulletin of the London Mathematical Society, 30(6):643–650, 1998

Grigor’yan, Alexander and Noguchi, Masakazu. The heat kernel on hyperbolic space.Bulletin of the London Mathematical Society, 30(6):643–650, 1998. 41

work page 1998

[14] [14]

Logarithmic Sobolev inequalities and stochas- tic Ising models.Journal of Statistical Physics, 46(5-6):1159–1194, 1987

Holley, Richard and Stroock, Daniel. Logarithmic Sobolev inequalities and stochas- tic Ising models.Journal of Statistical Physics, 46(5-6):1159–1194, 1987

work page 1987

[15] [15]

Heat flows on hyperbolic spaces.Journal of Differential Geometry, 108(3):495 – 529, 2018

Lemm, Marius and Markovic, Vladimir. Heat flows on hyperbolic spaces.Journal of Differential Geometry, 108(3):495 – 529, 2018

work page 2018

[16] [16]

Lp asymptotics for the heat equation on symmetric spaces for non-symmetric solutions.International Mathematics Research Notices, 2025(7):26, 2025

Papageorgiou, Effie. Lp asymptotics for the heat equation on symmetric spaces for non-symmetric solutions.International Mathematics Research Notices, 2025(7):26, 2025

work page 2025

[17] [17]

Springer, Cham, third edition, 2016

Petersen, Peter.Riemannian Geometry, volume 171 ofGraduate Texts in Mathe- matics. Springer, Cham, third edition, 2016

work page 2016

[18] [18]

Analysis of the laplacian on the complete riemannian manifold

Strichartz, Robert. Analysis of the laplacian on the complete riemannian manifold. Journal of Functional Analysis, 52:48–79, 1983

work page 1983

[19] [19]

Kinetic approach to the asymptotic behaviour of the solution to diffusion equations.Rend

Toscani, Giuseppe. Kinetic approach to the asymptotic behaviour of the solution to diffusion equations.Rend. Sem. Mat. Univ. Padova, 16:329–346, 1996

work page 1996

[20] [20]

Entropy production and the rate of convergence to equilibrium for the fokker-planck equation.Quarterly of Applied Mathematics, 57(3):521–541, 1999

Toscani, Giuseppe. Entropy production and the rate of convergence to equilibrium for the fokker-planck equation.Quarterly of Applied Mathematics, 57(3):521–541, 1999

work page 1999

[21] [21]

Asymptotic behaviour for the heat equation in hyperbolic space.Communications in Partial Differential Equations, 17(7-8):1263–1279, 1992

Vázquez, Juan Luis. Asymptotic behaviour for the heat equation in hyperbolic space.Communications in Partial Differential Equations, 17(7-8):1263–1279, 1992

work page 1992

[22] [22]

L.Asymptotic behavior methods for the Heat Equation

Vázquez, Juan Luis. Asymptotic behaviour methods for the heat equation. con- vergence to the gaussian.arXiv:1706.10034, 2018. José A. Cañizo. Departamento de Matemática Aplicada & IMAG, Universidad de Granada. A vda. Fuentenuev a S/N, 18071 Granada, Spain. E-mail address:canizo@ugr.es Alejandro Gárriz. Departamento de Matemática Aplicada & IMAG, Universid...

work page arXiv 2018