  Recent observations by the Large Area Telescope (LAT) onboard the Fermi
satellite have revealed bright gamma-ray emission from middle-aged supernova
remnants (SNRs) inside our Galaxy. These remnants, which also possess bright
non-thermal radio shells, are often found to be interacting directly with
surrounding gas clouds. We explore the non-thermal emission mechanism at these
dynamically evolved SNRs by constructing a hydrodynamical model. Two scenarios
of particle acceleration, either a re-acceleration of Galactic cosmic rays
(CRs) or an efficient nonlinear diffusive shock acceleration (NLDSA) of
particles injected from downstream, are considered. Using parameters inferred
from observations, our models are contrasted with the observed spectra of SNR
W44. For the re-acceleration case, we predict a significant enhancement of
radio and GeV emission as the SNR undergoes a transition into the radiative
phase. If sufficiently strong magnetic turbulence is present in the molecular
cloud, the re-acceleration scenario can explain the observed broadband spectral
properties. The NLDSA scenario also succeeds in explaining the $\gamma$-ray
spectrum but fails to reproduce the radio spectral index. Efficient NLDSA also
results in a significant post-shock non-thermal pressure that limits the
compression during cooling and prevents the formation of a prominent dense
shell. Some other interesting differences between the two models in
hydrodynamical behavior and resulting spectral features are illustrated.
