  We have calculated the temperature profiles for phonon heat transport in a
suspended membrane with a radially symmetric heat source in the two extreme
cases of either fully ballistic or fully diffusive transport. Theoretical
results confirm that it is possible to distinguish these two transport
mechanisms from the radial temperature profiles alone. Models are also compared
to experimental data measured with 40 nm thick, free standing silicon nitride
membranes below 1 K by using tunnel junction (SINIS) thermometers. The measured
temperature profile is qualitatively in agreement with the ballistic model.
