Stabilization for equations of one-dimensional viscous compressible heat-conducting media with nonmonotone equation of state

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)=p0(u)+p1(u)θ linear with respect to the temperature θ, but we admit rather general nonmonotone functions p0 and p1 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 Lq for u, in L2 for θ, and in Lq for v (the velocity), for any q [2,∞). For completeness, the proof of the corresponding global existence theorem is also included.