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

1. Problem Statement

The paper investigates the large-time asymptotic behavior of positive solutions to the Heat Equation on the hyperbolic space $\mathbb{H}^d$ ($d \ge 2$):
$$\begin{cases} u_t = \Delta_{\mathbb{H}^d} u, & x \in \mathbb{H}^d, \, t > 0, \\ u(0, \cdot) = u_0, & x \in \mathbb{H}^d. \end{cases}$$
The primary challenge addressed is that, unlike in Euclidean space $\mathbb{R}^d$, solutions in hyperbolic space do not universally converge to a single fundamental solution (Gaussian profile) centered at the origin or the center of mass. Instead, the asymptotic profile depends heavily on the geometry of the space and the specific distribution of the initial mass.

Previous works (e.g., Vázquez, Papageorgiou) established convergence to specific profiles but lacked quantitative rates of convergence (speed) and often relied on geometric harmonic analysis techniques that were distinct from standard PDE methods. This paper aims to:

1. Identify precise asymptotic profiles for radial, horospheric, and general integrable initial data.
2. Provide sharp convergence rates in $L^1$ and $L^\infty$ norms.
3. Develop a unified framework using entropy estimates adapted to non-compact Riemannian manifolds with non-trivial curvature.

2. Methodology

The core methodology involves adapting entropy methods (specifically relative entropy and logarithmic Sobolev inequalities) to the geometry of $\mathbb{H}^d$.

A. Coordinate Systems and Change of Variables

The authors utilize specific coordinate systems to capture the symmetries of the solutions:

• Polar (Elliptic) Coordinates: For radial solutions, $x = (r, \theta)$, where $r$ is the geodesic distance from a pole $O$.
• Parabolic (Horospheric) Coordinates: For horospheric solutions, $x = (r, y)$, where $r$ is the signed distance to a horosphere $H$.

To analyze the large-time behavior, they introduce a time-dependent change of variables that tracks the "bulk" of the mass, which in $\mathbb{H}^d$ drifts linearly with time due to the negative curvature:

• Time scaling: $t(\tau) = e^\tau - 1$.
• Spatial scaling: $r(\tau, \rho) = (d-1)e^\tau + e^{\tau/2}\rho$.
• Function transformation: $v(\tau, \rho, \theta) = e^{\tau/2} \sinh^{d-1}(r) u(t, r, \theta)$.

This transformation maps the problem from the hyperbolic geometry to a setting resembling a Fokker-Planck equation on $\mathbb{R}$, allowing the use of standard entropy techniques.

B. Entropy Functional and Transient Equilibria

The authors define a relative entropy functional:
$$H(\tau) = \int v \ln\left(\frac{v}{F}\right) d\mu,$$
where $F$ is a transient equilibrium profile. Unlike classical entropy methods where the equilibrium is static, here $F$ is time-dependent and incorporates the geometric information of the manifold (e.g., the volume form $\sinh^{d-1}(r)$).

Key steps in the proof include:

1. Deriving the Evolution Equation: Showing that the transformed function $v$ satisfies a parabolic equation with a drift term.
2. Entropy Decay: Proving that $\partial_\tau H(\tau) \le -\lambda H(\tau) + \text{error terms}$ using Logarithmic Sobolev inequalities (specifically adapted to the hyperbolic weight).
3. Csiszár-Kullback Inequality: Using the decay of entropy to bound the $L^1$ distance between the solution and the asymptotic profile.
4. Smoothing Effects: Utilizing $L^1-L^\infty$ regularizing effects to derive convergence rates in the $L^\infty$ norm.

3. Key Contributions and Results

A. Radially Symmetric Solutions

For initial data $u_0$ depending only on the distance to a pole $O$:

• Profile: The solution converges to a profile proportional to the fundamental solution $P_d(t, x)$.
• Rate: The authors prove a convergence rate of $O(t^{-1/2})$ in $L^1$ and $O(t^{-2}e^{-\lambda_1 t})$ in $L^\infty$, where $\lambda_1 = (d-1)^2/4$ is the bottom of the spectrum of the Laplacian.
• Significance: This improves upon previous results by providing explicit speeds and identifying the precise time-dependent coefficient $C(t)$ that converges to a constant.

B. Horospheric Solutions

For initial data depending only on the distance to a horosphere:

• Profile: The solution converges to a Gaussian profile shifted by the drift $(d-1)t$.
• Rate: Convergence is established at $O(t^{-1/2})$ in $L^1$ and $O(t^{-1})$ in $L^\infty$.
• Note: These solutions are generally not in $L^1(\mathbb{H}^d)$ but have a conserved "horospheric mass."

C. General Non-Symmetric Solutions (Main Contribution)

For general $u_0 \in L^1(\mathbb{H}^d)$, the solution does not converge to a single fundamental solution. Instead, it converges to a profile that retains a "memory" of the initial distribution's angular components.

• The Asymptotic Profile:
  $$u(t, x) \approx \Phi(\theta_x) P_d(t, x)$$
  where $\Phi(\theta_x)$ is a memory function defined by:
  $$\Phi(\theta_x) = \int_{\mathbb{H}^d} u_0(y) \left[ \cosh(r_y) - \sinh(r_y) \langle \theta_x, \theta_y \rangle \right]^{-(d-1)} d\mu(y).$$
  This function $\Phi$ captures the directional mass distribution of the initial data.

• Convergence Rates:
  Under a mild growth condition on the initial data (an $\epsilon$-moment condition, satisfied by compactly supported data), the authors prove:
  $$\| u(t, \cdot) - \Phi(\cdot) P_d(t, \cdot) \|_{L^1(\mathbb{H}^d)} \le C t^{-1/2}.$$
  This is the first result to provide a sharp convergence rate for general non-symmetric solutions in hyperbolic space.

4. Significance and Impact

1. Unified Framework: The paper successfully unifies the study of radial, horospheric, and general solutions under a single entropy-based framework, avoiding the disparate techniques of geometric harmonic analysis used in prior works (e.g., Papageorgiou).
2. Sharp Rates: It resolves the lack of convergence speeds in previous literature, providing explicit polynomial decay rates ($t^{-1/2}$) for the $L^1$ error.
3. Geometric Adaptation: It demonstrates how entropy methods can be rigorously adapted to non-compact Riemannian manifolds with non-trivial curvature, specifically handling the volume form and drift terms inherent to hyperbolic geometry.
4. Memory of Initial Data: The work clarifies the non-universal nature of asymptotic profiles in hyperbolic space. Unlike Euclidean space where the profile is a Gaussian centered at the center of mass, in $\mathbb{H}^d$, the profile is a superposition of heat kernels weighted by the initial angular distribution, a phenomenon quantified by the function $\Phi$.

5. Conclusion

The paper establishes that while the heat equation in hyperbolic space exhibits complex asymptotic behavior due to the geometry, this behavior is predictable and quantifiable. By employing adapted entropy estimates, the authors provide a complete description of the long-time dynamics, identifying the specific "memory" function that dictates the asymptotic shape and proving that the convergence to this shape occurs at a rate of $t^{-1/2}$ in the $L^1$ norm. This represents a significant advancement in the understanding of diffusion processes on non-compact manifolds.