On the $n$-dimensional real hyperbolic space, let $\Delta$ be its nonnegative Laplace--Beltrami operator whose bottom of the spectrum is $\lambda_{0}$, and let $\sigma \in (0,1)$. In this talk, on the one hand, we discuss the Fujita exponent for the fractional heat equation \[ \partial_t u + \Delta^{\sigma}u = e^{\beta t}|u|^{\gamma-1}u, \] by proving that nontrivial positive global solutions exist if and only if $\gamma\geq 1 + \beta/ \lambda_{0}^{\sigma}$. On the other hand, we prove the existence of non-negative, bounded and finite energy solutions of the semilinear fractional elliptic equation \[ \Delta^{\sigma} v - \lambda^{\sigma} v - v^{\gamma}=0 \] for $0\leq \lambda \leq \lambda_{0}$ and $1'<\gamma'< \frac{n+2\sigma}{n-2\sigma}$. The latter problem, aside from its independent interest, is actually instrumental to the former. At the core of the results above 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 an associated Rellich--Kondrachov-like compact embedding theorem for radial functions.
Joint work with T. Bruno (Genova).
