  Recent gamma-ray observations show that middle aged supernova remnants (SNRs)
interacting with molecular clouds (MCs) can be sources of both GeV and TeV
emission. Based on the MC association, two scenarios have been proposed to
explain the observed gamma-ray emission. In one, energetic cosmic ray (CR)
particles escape from the SNR and then illuminate nearby MCs, producing
gamma-ray emission, while the other involves direct interaction between the SNR
and MC. In the direct interaction scenario, re-acceleration of pre-existing CRs
in the ambient medium is investigated while particles injected from the thermal
pool are neglected in view of the slow shock speeds in middle aged SNRs.
However, standard diffusive shock acceleration (DSA) theory produces a steady
state particle spectrum that is too flat compared to observations, which
suggests that the high energy part of the observed spectrum has not yet reached
a steady state. We derive a time dependent DSA solution in the test particle
limit for re-acceleration of pre-existing CRs case and show that it is capable
of reproducing the observed gamma-ray emission in SNRs like IC 443 and W44, in
the context of a MC interaction model. We also provide a simple physical
picture to understand the time dependent DSA spectrum. A spatially averaged
diffusion coefficient around the SNR can be estimated through fitting the
gamma-ray spectrum. The spatially averaged diffusion coefficient in middle aged
SNRs like IC 443 and W44 is estimated to be ~ 10^25 cm^2/s at ~ 1GeV, which is
between the Bohm limit and interstellar value.
