  In this work, we study the propagation of scalar fields in the gravitational
background of a higher-dimensional Schwarzschild-de-Sitter black hole as well
as on the projected-on-the-brane 4-dimensional background. The scalar fields
have also a non-minimal coupling to the corresponding, bulk or brane, scalar
curvature. We perform a comprehensive study by deriving exact numerical results
for the greybody factors, and study their profile in terms of particle and
spacetime properties. We then proceed to derive the Hawking radiation spectra
for a higher-dimensional Schwarzschild-de-Sitter black hole, and we study both
bulk and brane channels. We demonstrate that the non-minimal field coupling,
that creates an effective mass term for the fields, suppresses the energy
emission rates while the cosmological constant assumes a dual role. By
computing the relative energy rates and the total emissivity ratio for bulk and
brane emission, we demonstrate that the combined effect of a large number of
extra dimensions and value of the field coupling gives to the bulk channel the
clear domination in the bulk-brane energy balance.
