  The common approach to compute the cosmic-ray distribution in an starburst
galaxy or region is equivalent to assume that at any point within that
environment, there is an accelerator inputing cosmic rays at a reduced rate.
This rate should be compatible with the overall volume-average injection, given
by the total number of accelerators that were active during the starburst age.
These assumptions seem reasonable, especially under the supposition of an
homogeneous and isotropic distribution of accelerators. However, in this
approach the temporal evolution of the superposed spectrum is not explicitly
derived; rather, it is essentially assumed ab-initio. Here, we test the
validity of this approach by following the temporal evolution and spatial
distribution of the superposed cosmic-ray spectrum and compare our results with
those from theoretical models that treat the starburst region as a single
source. In the calorimetric limit (with no cosmic-ray advection), homogeneity
is reached (typically within 20%) across most of the starburst region. However,
values of center-to-edge intensity ratios can amount to a factor of several.
Differences between the common homogeneous assumption for the cosmic-ray
distribution and our models are larger in the case of two-zone geometries, such
as a central nucleus with a surrounding disc. We have also found that the decay
of the cosmic-ray density following the duration of the starburst process is
slow, and even approximately 1 Myr after the burst ends (for a gas density of
35 cm-3) it may still be within an order of magnitude of its peak value. Based
on our simulations, it seems that the detection of a relatively hard spectrum
up to the highest gamma-ray energies from nearby starburst galaxies favors a
relatively small diffusion coefficient (i.e., long diffusion time) in the
region where most of the emission originates.
