  Several supernovae (SNe) with an unusually dense circumstellar medium (CSM)
have been recently observed at radio frequencies. Their radio emission is
powered by relativistic electrons that can be either accelerated at the SN
shock (primaries) or injected as a by-product (secondaries) of inelastic
proton-proton collisions. We investigate the radio signatures from secondary
electrons, by detailing a semi-analytical model to calculate the temporal
evolution of the distributions of protons, primary and secondary electrons.
With our formalism, we track the cooling history of all the particles that have
been injected into the emission region up to a given time, and calculate the
resulting radio spectra and light curves. For a SN shock propagating through
the progenitor wind, we find that secondary electrons control the early radio
signatures, but their contribution decays faster than that of primary
electrons. This results in a flattening of the light curve at a given radio
frequency that depends only upon the radial profiles of the CSM density and of
the shock velocity, $v_0$. The relevant transition time at the peak frequency
is $\sim 190 \, K_{\rm ep,-3}^{-1} A_{\rm w, 16} \beta_{0, -1.5}^{-2}\,{\rm
d}$, where $A_{\rm w}$ is the wind mass-loading parameter, $\beta_0=v_0/c$ and
$K_{\rm ep}$ is the electron-to-proton ratio of accelerated particles. We
explicitly show that late peak times at 5 GHz (i.e., $t_{\rm pk} \gtrsim
300-1000$ d) suggest a shock wave propagating in a dense wind ($A_{\rm w}
\gtrsim 10^{16}-10^{17}$ gr cm$^{-1}$), where secondary electrons are likely to
power the observed peak emission.
