  This paper concerns the physical behaviors of any solutions to the one
dimensional compressible Navier-Stokes equations for viscous and heat
conductive gases with constant viscosities and heat conductivity for fast
decaying density at far fields only. First, it is shown that the specific
entropy becomes not uniformly bounded immediately after the initial time, as
long as the initial density decays to vacuum at the far field at the rate not
slower than $O\left(\frac1{|x|^{\ell_\rho}}\right)$ with $\ell_\rho>2$.
Furthermore, for faster decaying initial density, i.e., $\ell_\rho\geq4$, a
sharper result is discovered that the absolute temperature becomes uniformly
positive at each positive time, no matter whether it is uniformly positive or
not initially, and consequently the corresponding entropy behaves as
$O(-\log(\varrho_0(x)))$ at each positive time, independent of the boundedness
of the initial entropy. Such phenomena are in sharp contrast to the case with
slowly decaying initial density of the rate no faster than $O(\frac1{x^2})$,
for which our previous works \cite{LIXINADV,LIXINCPAM,LIXIN3DK} show that the
uniform boundedness of the entropy can be propagated for all positive time and
thus the temperature decays to zero at the far field. These give a complete
answer to the problem concerning the propagation of uniform boundedness of the
entropy for the heat conductive ideal gases and, in particular, show that the
algebraic decay rate $2$ of the initial density at the far field is sharp for
the uniform boundedness of the entropy. The tools to prove our main results are
based on some scaling transforms, including the Kelvin transform, and a Hopf
type lemma for a class of degenerate equations with possible unbounded
coefficients.
