We study the non-thermal emission from old shell-type supernova remnants (SNRs) on the frame of a time-dependent model. In this model, the time-dependent non-thermal spectra of both primary electrons and protons as well as secondary electron/positron ($e^{pm}$) pairs can be calculated numerically by taking into account the evolution of the secondary $e^{pm}$ pairs produced from proton-proton (p-p) interactions due to the accelerated protons collide with the ambient matter in an SNR. The multi-wavelength photon spectrum for a given SNR can be produced through leptonic processes such as electron/positron synchrotron radiation, bremsstrahlung and inverse Compton scattering as well as hadronic interaction. Our results indicate that the non-thermal emission of the secondary $e^{pm}$ pairs is becoming more and more prominent when the SNR ages in the radiative phase because the source of the primary electrons has been cut off and the electron synchrotron energy loss is significant for a radiative SNR, whereas the secondary $e^{pm}$ pairs can be produced continuously for a long time in the phase due to the large energy loss time for the p-p interaction. We apply the model to two old SNRs, G8.7$-$0.1 and G23.3$-$0.3, and the predicted results can explain the observed multi-wavelength photon spectra for the two sources.
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