Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917416540307456 |
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| author | Bruno, Tommaso Papageorgiou, Effie |
| author_facet | Bruno, Tommaso Papageorgiou, Effie |
| contents | Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $λ_{0}$, and $σ\in (0,1)$.
The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u,\] by proving that nontrivial positive global solutions exist if and only if $γ\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[
Δ^σ v - λ^σ v - v^γ=0 \] for $0\leq λ\leq λ_{0}$ and $1<γ< \frac{n+2σ}{n-2σ}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former.
\smallskip
At the core of our results stands a novel fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12349 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces Bruno, Tommaso Papageorgiou, Effie Analysis of PDEs 35J60, 26A33, 43A85, 22E30 Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $λ_{0}$, and $σ\in (0,1)$. The aim of this paper is twofold. On the one hand, we determine the Fujita exponent for the fractional heat equation \[\partial_{t} u + Δ^σu = e^{βt}|u|^{γ-1}u,\] by proving that nontrivial positive global solutions exist if and only if $γ\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[ Δ^σ v - λ^σ v - v^γ=0 \] for $0\leq λ\leq λ_{0}$ and $1<γ< \frac{n+2σ}{n-2σ}$. The two problems are known to be connected and the latter, aside from its independent interest, is actually instrumental to the former. \smallskip At the core of our results stands a novel fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$ fractional Sobolev spaces, which sharpens those known so far, and which holds more generally on Riemannian symmetric spaces of non-compact type. We also establish an associated Rellich--Kondrachov-like compact embedding theorem for radial functions, along with other related properties. |
| title | Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian on hyperbolic spaces |
| topic | Analysis of PDEs 35J60, 26A33, 43A85, 22E30 |
| url | https://arxiv.org/abs/2509.12349 |