Asymptotic estimates for a semigroup related to compressible viscous fluid flow

The paper is related to the question of stability for the motionless spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. It considers a perturbation of the linearization of the governing equations of this problem, taking a step in the derivation of estimates which will allow us to prove non-linear stability of the equilibria. The perturbed operator, like the linearization considered earlier, generates an analytic semigroup, which allows us to derive asymptotic estimates as