  In evolved supernova remnants (SNRs) interacting with molecular clouds, such
as IC 443, W44, and 3C391, a highly inhomogeneous structure consisting of a
forward shock of moderate Mach number, a cooling layer, a dense radiative shell
and an interior region filled with hot tenuous plasma is expected. We present a
kinetic model of nonthermal electron injection, acceleration and propagation in
that environment and find that these SNRs are efficient electron accelerators
and sources of hard X- and gamma-ray emission. The energy spectrum of the
nonthermal electrons is shaped by the joint action of first and second order
Fermi acceleration in a turbulent plasma with substantial Coulomb losses.
Bremsstrahlung, synchrotron, and inverse Compton radiation of the nonthermal
electrons produce multiwavelength photon spectra in quantitative agreement with
the radio and the hard emission observed by ASCA and EGRET from IC 443. We
distinguish interclump shock wave emission from molecular clump shock wave
emission accounting for a complex structure of molecular cloud. Spatially
resolved X- and gamma- ray spectra from the supernova remnants IC 443, W44, and
3C391 as might be observed with BeppoSAX, Chandra XRO, XMM, INTEGRAL and GLAST
would distinguish the contribution of the energetic lepton component to the
gamma-rays observed by EGRET.
