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This paper concerns the physical behaviors of any solutions to the onedimensional compressible Navier-Stokes equations for viscous and heatconductive gases with constant viscosities and heat conductivity for fastdecaying density at far fields only. First, it is shown that the specificentropy becomes not uniformly bounded immediately after the initial time, aslong as the initial density decays to vacuum at the far field at the rate notslower than $O\left(\frac1{|x|^{\ell_\rho}}\right)$ with $\ell_\rho>2$.Furthermore, for faster decaying initial density, i.e., $\ell_\rho\geq4$, asharper result is discovered that the absolute temperature becomes uniformlypositive at each positive time, no matter whether it is uniformly positive ornot initially, and consequently the corresponding entropy behaves as$O(-\log(\varrho_0(x)))$ at each positive time, independent of the boundednessof the initial entropy. Such phenomena are in sharp contrast to the case withslowly decaying initial density of the rate no faster than $O(\frac1{x^2})$,for which our previous works \cite{LIXINADV,LIXINCPAM,LIXIN3DK} show that theuniform boundedness of the entropy can be propagated for all positive time andthus the temperature decays to zero at the far field. These give a completeanswer to the problem concerning the propagation of uniform boundedness of theentropy for the heat conductive ideal gases and, in particular, show that thealgebraic decay rate $2$ of the initial density at the far field is sharp forthe uniform boundedness of the entropy. The tools to prove our main results arebased on some scaling transforms, including the Kelvin transform, and a Hopftype lemma for a class of degenerate equations with possible unboundedcoefficients.

Instantaneous unboundedness of the entropy and uniform positivity of the  temperature for the compressible Navier-Stokes equations with fast decay  density