Molecular clouds are known to be clumpy, with dense molecular clumps occupying only a few percent of the volume. A supernova remnant then evolves primarily in the interclump medium, and becomes radiative at a radius of about 6 pc, forming a shell that is magnetically supported. When this shell interacts with the dense clumps, the molecular shock fronts are driven by a considerable overpressure compared to the pressure in the rest of the remnant. Observations of the remnants W44 and IC 443 can be understood in this model. Both remnants are shell sources of radio synchrotron emission, which can be attributed to relativistic electrons in the cool radiative shell. If ambient cosmic ray electrons are accelerated by the shock front and by the postshock compression, the radio fluxes and the flat spectral indices of W44 and IC 443 can be explained, as well as the high energy gamma ray emission from bremsstrahlung.

31 pages, 4 figures, revised, in press, ApJ, Feb. 1, 1999