  We develop the theory of heat conductivity in supported graphene, accounting
for coherent phonon scattering on disorder induced by an amorphous substrate.
We derive spectra for in-plane and out-of-plane phonons in the framework of
Green's function approach. The energetic parameters of the theory are obtained
using the molecular dynamics simulations for graphene on SiO$_2$ substrate. The
heat conductivity is calculated by the Boltzmann transport equation. We find
that the interaction with the substrate drastically reduces the phonon lifetime
and completely suppresses the contribution of ZA phonons to the heat
conductivity. As a result, the total heat conductivity is reduced by several
times, which matches with the tendency observed for available experimental
data. The considered effect is important for managing thermal properties of the
graphene-based electronic devices.
