We perform hydrodynamical simulations of the interaction between supernova (SN) ejecta and circumstellar medium (CSM) for SN 1993J and SN 2011dh, and calculate the radio and X-ray emissions expected from the shocked gas at late epochs (t). Considering the ejecta structure from multi-group radiation hydrodynamics simulation, we find that the observed rapid drop in radio and X-ray light curves of SN 1993J at t > 3000 days may be due to a change in the mass-loss rate ( ) ∼6500 yr prior to the explosion of the SN. The exact epoch scales inversely with the assumed wind velocity of vw = 10 . The progenitor of this SN very likely belonged to a binary system, where, during its evolution, the primary had transferred material to the secondary. It is argued in this paper that the change in can happen because of a change in the mass accretion efficiency ( ) of the companion star. It is possible that before ∼6500 (vw/10 )−1 yr prior to the explosion, was high, and thus the CSM was tenuous, which causes the late-time downturn in fluxes. In the case of SN 2011dh, the late-time evolution is found to be consistent with a wind medium with /vw = 4 × 10−6 /10 . It is difficult from our analysis to predict whether the progenitor of this SN had a binary companion; however, if future observations show a similar decrease in radio and X-ray fluxes, then this would give strong support to a scenario where both SNe had undergone a similar kind of binary evolution before explosion.