Blow-up for semilinear parabolic equations in cones of the hyperbolic space

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{\mu t}{u}^p$ ($\mu \in \mathbb{R},p>1$), posed on cones of the hyperbolic space. Under a certain assumption on $\mu$ and $p$, related to the bottom of the spectrum of $-\mathrm{\Delta }$ in ${\mathbb{H}}^n$, we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters $\mu$ and $p$ satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters $\mu$ and $p$ are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting, and it is essentially due to a specific geometric feature of ${\mathbb{H}}^n$.