  The evolution of the radio emission of shell-type Supernova remnants (SNRs)
is modeled within the framework of the simple and commonly used assumptions
that the mechanism of diffusive shock acceleration (DSA) is responsible for
generating radio emitting electrons and that the magnetic field is the typical
interstellar field compressed at the shock. It is considered that electrons are
injected into the mechanism in test-particle regime directly from the high
energy tail of the downstream Maxwellian distribution function. The model can
be applied to most of the observed SNRs. It is shown that the model
successfully explains the many averaged observational properties of evolved
shell-type SNRs. In particular, the radio surface brightness ($\Sigma$) evolves
with diameter as $\sim D^{-(0.3 \div 0.5)}$, while the bounding shock is strong
(Mach number is ${\mathcal M} \geq10$), followed by steep decrease (steeper
than $\sim D^{-4.5}$) for ${\cal M} <10$. Such evolution of the surface
brightness with diameter and its strong dependence on the environmental
parameters strongly reduce the usefulness of $\Sigma - D$ relations as a tool
for determining the distances to SNRs. The model predicts no radio emission
from SNRs in the late radiative stage of evolution and the existence of
radio-quiet but relatively active SNRs is possible. Our model easily explains
very large-diameter radio sources such as the Galactic Loops and the candidates
for Hypernova radio remnants. The model predicts that most of the observed SNRs
are located in a tenuous phase of the ISM. From the comparison of the model
results with the statistics of evolved shell-type SNRs, we were able to
estimate the fraction of electrons accelerated from the thermal pool in the
range $(3\div 11) \times 10^{- 4}$.
