We consider the Navier-Stokes system describing motions of viscous compressible heat-conducting and ''self-gravitating'' media. We use the state function of the form p(u,θ)=p 0 (u)+p 1 (u)θ linear with respect to the temperature θ, but we admit rather general nonmonotone functions p 0 and p 1 of u, which allows us to treat various physical models of nuclear fluids (for which p and u are the pressure and the specific volume) or thermoviscoelastic solids. For solutions to an associated initial-boundary value problem with ''fixed-free'' boundary conditions and arbitrarily large data, we prove a collection of estimates independent of time interval, including uniform two-sided bounds for u, and describe asymptotic behavior as t->~. Namely, we establish the stabilization pointwisely and in L q for u, in L 2 for θ, and in L q for v (the velocity), for any q [2,~). For completeness, the proof of the corresponding global existence theorem is also included.