Inequalities for radial solutions to the wave equation on the hyperbolic space

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $Δ_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(Δ_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0