Mathematical analysis of unstable density fluctuations in the dissipative gravitational collapse

A detailed analysis of the dynamics of unstable modes present in the linearized Navier-Stokes-Fourier system in the presence of a gravitational field is carried out. The transition between the non-dissipative and dissipative regimes is explored from the mathematical point of view by considering the same equation of state in both cases, a procedure which sheds light on the discontinuity present in the critical parameter. It is also shown that two distinct behaviours for the propagation of unstable modes can be identified below the critical value for a gravitational instability to occur, depending on a parameter $r$ that quantifies the relevance of dissipation relative to gravitational effects.