  The processes of the coronal plasma heating and cooling were previously shown
to significantly affect the dynamics of slow magnetoacoustic (MA) waves,
causing amplification or attenuation, and also dispersion. However, the entropy
mode is also excited in such a thermodynamically active plasma and is affected
by the heating/cooling misbalance too. This mode is usually associated with the
phenomenon of coronal rain and formation of prominences. Unlike the adiabatic
plasmas, the properties and evolution of slow MA and entropy waves in
continuously heated and cooling plasmas get mixed. Different regimes of the
misbalance lead to a variety of scenarios for the initial perturbation to
evolve. In order to describe properties and evolution of slow MA and entropy
waves in various regimes of the misbalance, we obtained an exact analytical
solution of the linear evolutionary equation. Using the characteristic
timescales and the obtained exact solution, we identified regimes with
qualitatively different behaviour of slow MA and entropy modes. For some of
those regimes, the spatio-temporal evolution of the initial Gaussian pulse is
shown. In particular, it is shown that slow MA modes may have a range of
non-propagating harmonics. In this regime, perturbations caused by slow MA and
entropy modes in a low-$\beta$ plasma would look identically in observations,
as non-propagating disturbances of the plasma density (and temperature) either
growing or decaying with time. We also showed that the partition of the initial
energy between slow MA and entropy modes depends on the properties of the
heating and cooling processes involved. The obtained exact analytical solution
could be further applied to the interpretation of observations and results of
numerical modelling of slow MA waves in the corona and the formation and
evolution of coronal rain.
