  The evolutionary behavior and multi-wavelength emission properties of bubbles
around galaxies, such as the \textit{Fermi} bubbles of the Milky Way, is
unsettled. We perform 3D magnetohydrodynamical simulations to investigate the
evolution of leptonic galaxy bubbles driven by a 0.3 Myr intense explosive
outburst from the nucleus of Milky Way-like galaxies. Adopting an ageing model
for their leptonic cosmic rays, we post-process our simulations to compute the
multi-wavelength emission properties of these bubbles. We calculate the
resulting spectra emitted from the bubbles from radio frequencies to
$\gamma$-rays, and construct emission maps in four energy bands to show the
development of the spatial emission structure of the bubbles. The simulated
bubbles show a progression in their spectral properties as they age. In
particular, the TeV $\gamma$-ray emission is initially strong and dominated by
inverse Compton scattering, but falls rapidly after $\sim$ 1 Myr. By contrast,
the radio synchrotron emission remains relatively stable and fades slowly over
the lifetime of the bubble. Based on the emission properties of our
post-processed simulations, we demonstrate that $\gamma$-ray observations will
be limited in their ability to detect galaxy bubbles, with only young bubbles
around nearby galaxies being within reach. However, radio observations with
e.g. the up-coming Square Kilometer Array, would be able to detect
substantially older bubbles at much greater distances, and would be better
placed to capture the evolutionary progression and diversity of galaxy bubble
populations.
