Global strong solutions to a one-dimensional full non-Newtonian fluid with far field vacuum

In this paper, the Cauchy problem for a one-dimensional heat conducting compressible non-Newtonian fluid is considered. The constitute equation of the non-Newtonian fluid is determined by two nonlinear terms $(\mathrm{\mid }{u}_x{\mathrm{\mid }}^{q-2}{u}_x{)}_x$ and $(\mathrm{\mid }{\theta }_x{\mathrm{\mid }}^{p-2}{\theta }_x{)}_x$ with $1<p,q<2.$ When the vacuum occurs at the far field, the local and global existence of strong solutions are established for the Cauchy problem. The results indicate that the non-Newtonian fluid possesses time-dependence boundedness of the energy if the vacuum occurs at the far field and the density decays slowly at the far field as time goes to infinity. This is the key difference from the well-known result developed by Li and Xin (Advances in Mathematics 361(2020), 106923) for the one-dimensional heat conductive compressible Navier-Stokes system.