  In present study I examine the capability of diffusive shock acceleration
mechanism to explain existing data on radio emission from evolved large
diameter shell-type adiabatic supernova remnants (SNRs). Time-dependent
''onion-shell'' model for the radio emission of SNRs is developed, which is
based on the assumptions: a) acceleration takes place from thermal energies and
test-particle approximation is valid; b) the problem of injection is avoided by
introducing, like Bell (1978), two injection parameters; c) to take into
consideration very late stages of SNR evolution the analytic approximation of
Cox and Andersen (1982) for the shell structure is used; c)no radiative
cooling. Constructed Surface Brightness - Diameter $(\Sigma -D)$ tracks are
compared with the empirical $\Sigma -D$ diagram. The main conclusion of the
study is that the DSA mechanism is capable of explaining all the statistics of
radio SNRs including very large diameter remnants and giant galactic loops.
