  This work presents a linear analytical calculation on the stability and
evolution of a compressible, viscous self-gravitating (SG) Keplerian disc with
both horizontal thermal diffusion and a constant cooling timescale when an
axisymmetric structure is present and freely evolving. The calculation makes
use of the shearing sheet model and is carried out for a range of cooling
times. Although the solutions to the inviscid problem with no cooling or
diffusion are well known, it is non-trivial to predict the effect caused by the
introduction of cooling and of small diffusivities; this work focuses on
perturbations of intermediate wavelengths, therefore representing an extension
to the classical stability analysis on thermal and viscous instabilities. For
density wave modes the analysis can be simplified by means of a regular
perturbation analysis; considering both shear and thermal diffusivities, the
system is found to be overstable for intermediate and long wavelengths for
values of the Toomre parameter $Q \lesssim 2$; a non-SG instability is also
detected for wavelengths $\gtrsim 18H$, where $H$ is the disc scale height, as
long as $\gamma \lesssim 1.305$. The regular perturbation analysis does not
however hold for the entropy and potential vorticity slow modes as their ideal
growth rates are degenerate. To understand their evolution, equations for the
axisymmetric structure's amplitudes in these two quantities are analytically
derived and their instability regions obtained. The instability appears boosted
by increasing the value of the adiabatic index and of the Prandtl number, while
it is quenched by efficient cooling.

Solar and Stellar Astrophysics;Earth and Planetary Astrophysics;Fluid Dynamics