  In this paper, we investigate the wellposedness of the non-isentropic
compressible Navier-Stokes/Allen-Cahn system with the heat-conductivity
proportional to a positive power of the temperature. This system describes the
flow of a two-phase immiscible heat-conducting viscous compressible mixture.
The phases are allowed to shrink or grow due to changes of density in the fluid
and incorporates their transport with the current. We established the global
existence and uniqueness of strong solutions for this system in 1-D, which
means no phase separation, vacuum, shock wave, mass or heat or phase
concentration will be developed in finite time, although the motion of the
two-phase immiscible flow has large oscillations and the interaction between
the hydrodynamic and phase-field effects is complex. Our result can be regarded
as a natural generalization of the Kazhikhov-Shelukhin's result
([Kazhikhov-Shelukhin. J. Appl. Math. Mech. 41 (1977)]) for the compressible
single-phase flow with constant heat conductivity to the non-isentropic
compressible immiscible two-phase flow with degenerate and nonlinear heat
conductivity.
