Can the morphology of gamma-ray emission distinguish annihilating from decaying dark matter?

The recent results from the PAMELA, ATIC, FERMI and HESS experiments have focused attention on the possible existence of high energy cosmic ray e^+ e^- that may originate from dark matter (DM) annihilations or decays in the Milky Way. Here we examine the morphology of the gamma-ray emission after propagation of the electrons generated by both annihilating and decaying dark matter models. We focus on photon energies of 1 GeV, 10 GeV, 50 GeV (relevant for the FERMI satellite) and consider different propagation parameters. Our main conclusion is that distinguishing annihilating from decaying dark matter may only be possible if the propagation parameters correspond to the most optimistic diffusion models. In addition, we point to examples where morphology can lead to an erroneous interpretation of the source injection energy.