Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces

1. Problem Statement and Context

The paper addresses two interconnected problems concerning nonlinear partial differential equations involving the fractional Laplace–Beltrami operator ($\Delta^\sigma$, with $\sigma \in (0,1)$) on the real hyperbolic space $\mathbb{H}^n$ ($n \ge 2$).

Problem A: The Fractional Fujita Problem
The authors investigate the global solvability of the fractional heat equation with an exponential time-dependent source:
$$\begin{cases} \partial_t u + \Delta^\sigma u = e^{\beta t}|u|^{\gamma-1}u, & t > 0, x \in \mathbb{H}^n \\ u(0, \cdot) = f \ge 0, \end{cases}$$
where $\beta > 0$ and $\gamma > 1$. The goal is to determine the Fujita exponent $\gamma^*$, which acts as a threshold separating finite-time blow-up from global existence for small initial data.

• Context: In Euclidean space $\mathbb{R}^n$, the Fujita exponent is $1 + 2\sigma/n$. On hyperbolic space with the standard Laplacian ($\sigma=1$), the spectrum bottom $\lambda_0 = (n-1)^2/4$ creates a spectral gap, altering the behavior. The paper extends this to the fractional case.

Problem B: The Semilinear Fractional Elliptic Equation
To solve the critical case of Problem A, the authors study the existence of solutions to the stationary equation:
$$\Delta^\sigma v - \lambda^\sigma v - v^\gamma = 0,$$
for $0 \le \lambda \le \lambda_0$ and $1 < \gamma < \frac{n+2\sigma}{n-2\sigma}$. This equation is instrumental in constructing supersolutions for the evolution problem.

2. Methodology and Technical Tools

The paper employs a sophisticated blend of harmonic analysis on symmetric spaces, functional analysis, and nonlinear PDE techniques.

A. Harmonic Analysis on $\mathbb{H}^n$

• Spectral Theory: Utilization of the Helgason–Fourier transform and spherical functions $\phi_\xi$. The spectrum of $\Delta$ is $[\lambda_0, \infty)$ with $\lambda_0 = (n-1)^2/4$.
• Heat Kernel Estimates: Precise pointwise bounds for the fractional heat kernel $P^\sigma_t(x)$ are derived via subordination to the standard heat kernel. These estimates distinguish between local behavior (near the origin) and asymptotic behavior (at infinity), crucial for handling the non-local operator.
• Kunze–Stein Phenomenon: Used to establish convolution inequalities on hyperbolic spaces, which are essential for proving boundedness of operators in $L^p$ spaces.

B. Functional Analysis and Sobolev Spaces

• New Scale of Sobolev Spaces: The authors define a Hilbert space $H_{\lambda, \sigma}$ as the completion of $C_c^\infty(\mathbb{H}^n)$ under the norm $\|u\|_{\lambda, \sigma} = \|(\Delta^\sigma - \lambda^\sigma)^{1/2}u\|_2$.
  • Crucially, when $\lambda = \lambda_0$, this space is strictly larger than the standard fractional Sobolev space $H^\sigma$, yet it retains useful properties like radialization invariance.
• Fractional Poincaré Inequality: A central technical achievement is proving a sharp inequality:
  $$\int_{\mathbb{H}^n} (|\Delta^{\sigma/2} \phi|^2 - \lambda_0^\sigma \phi^2) d\mu \ge C \|\phi\|_q^2$$
  for $2 < q \le \frac{2n}{n-2\sigma}$. This inequality is stronger than previously known fractional Poincaré inequalities on hyperbolic spaces.

C. Compactness and Regularity

• Radial Compactness: The authors prove a Rellich–Kondrachov-type compact embedding theorem for the subspace of radial functions in $H_{\lambda, \sigma}$ into $L^q$ for $q < \frac{2n}{n-2\sigma}$. This is achieved by combining local compactness with a decay estimate at infinity (using a hyperbolic analog of Lions' lemma).
• Regularity Bootstrapping: Once a weak solution is found, its regularity is upgraded to classical solutions ($C^{0, 2\sigma+\epsilon}$ or $C^{1, 2\sigma-1+\epsilon}$) using pointwise estimates of the convolution kernel of $(\Delta^\sigma - \lambda^\sigma)^{-1}$ and known regularity results for the fractional Laplacian.

3. Key Contributions and Results

Result 1: The Fujita Exponent (Theorem A)
The paper establishes the exact Fujita exponent for the fractional heat equation on $\mathbb{H}^n$:
$$\gamma^* = 1 + \frac{\beta}{\lambda_0^\sigma}$$

• Subcritical ($1 < \gamma < \gamma^*$): Every nontrivial nonnegative mild solution blows up in finite time.
• Supercritical ($\gamma > \gamma^*$): Global solutions exist for sufficiently small initial data.
• Critical ($\gamma = \gamma^*$):
  • If $\beta > \frac{2}{3}\lambda_0^\sigma$, global solutions exist for small data.
  • If $0 < \beta \le \frac{2}{3}\lambda_0^\sigma$, global solutions exist (proven via the elliptic equation).
  • Note: This differs from the Euclidean case where the critical exponent often leads to non-existence for all data. The spectral gap of $\mathbb{H}^n$ allows for global existence even in the critical regime under certain conditions.

Result 2: Existence for the Elliptic Equation (Theorem B)
For $0 \le \lambda \le \lambda_0$ and $1 < \gamma < \frac{n+2\sigma}{n-2\sigma}$, the equation $\Delta^\sigma v - \lambda^\sigma v - v^\gamma = 0$ admits at least one non-trivial, non-negative, radial, bounded solution with finite energy.

• The solution belongs to $L^q(\mathbb{H}^n)$ for all $q \in (2, \infty]$.
• The proof relies on the Nehari manifold method and the compact embedding of radial functions, overcoming the lack of compactness in the full space due to the non-compactness of $\mathbb{H}^n$.

Result 3: Fractional Poincaré Inequality (Theorem C)
The paper proves a novel inequality involving the shifted fractional Laplacian:
$$\int_{\mathbb{H}^n} (|\Delta^{\sigma/2} \phi|^2 - \lambda_0^\sigma \phi^2) d\mu \ge C \|\phi\|_q^2$$
This inequality is sharp and defines a new scale of Sobolev spaces. It is strictly stronger than the standard fractional Sobolev inequality involving $(\Delta - \lambda_0)^{\sigma/2}$.

4. Significance and Implications

• Completing the Picture: This work provides the first complete characterization of the Fujita phenomenon for fractional operators on hyperbolic spaces, resolving the critical case which was previously open or only partially understood.
• Spectral Gap Interaction: The results highlight how the spectral gap $\lambda_0$ of the hyperbolic Laplacian fundamentally alters the blow-up dynamics compared to Euclidean space. The exponential time factor $e^{\beta t}$ acts as a counterbalance to the negative curvature.
• Analytical Advances: The development of the new scale of Sobolev spaces $H_{\lambda_0, \sigma}$ and the associated sharp Poincaré inequality are significant contributions to analysis on non-compact symmetric spaces. These tools are likely to be applicable to other nonlinear problems on hyperbolic manifolds.
• Generalization: While the paper focuses on $\mathbb{H}^n$ (rank one), the authors demonstrate that the core results (specifically the Poincaré inequality and non-critical Fujita results) extend to general Riemannian symmetric spaces of non-compact type, with the critical exponent depending on the "pseudo-dimension" $\nu$ of the space.

In summary, the paper successfully bridges fractional calculus, spectral geometry, and nonlinear PDE theory to solve a long-standing problem regarding blow-up phenomena on hyperbolic spaces, introducing powerful new functional analytic tools in the process.