  The second-order Kuramoto equation describes synchronization of coupled
oscillators with inertia, which occur in power grids for example. Contrary to
the first-order Kuramoto equation it's synchronization transition behavior is
much less known. In case of Gaussian self-frequencies it is discontinuous, in
contrast to the continuous transition for the first-order Kuramoto equation.
Here we investigate this transition on large 2d and 3d lattices and provide
numerical evidence of hybrid phase transitions, that the oscillator phases
$\theta_i$, exhibit a crossover, while the frequency spread a real phase
transition in 3d. Thus a lower critical dimension $d_l^O=2$ is expected for the
frequencies and $d_l^R=4$ for the phases like in the massless case. We provide
numerical estimates for the critical exponents, finding that the frequency
spread decays as $\sim t^{-d/2}$ in case of aligned initial state of the phases
in agreement with the linear approximation. However in 3d, in the case of
initially random distribution of $\theta_i$, we find a faster decay,
characterized by $\sim t^{-1.8(1)}$ as the consequence of enhanced
nonlinearities which appear by the random phase fluctuations.
