Global strong/classical solutions to the one-dimensional compressible Navier-Stokes-Allen-Cahn system with density-dependent viscosity

This paper is concerned with a one-dimensional isentropic compressible Navier-Stokes-Allen-Cahn system density-dependent viscosity, which models the motion of mixture two viscous fluids. The case when pressure $ p(\rho) = \rho^\gamma $, viscosity \nu(\rho, \chi) \rho^\alpha interface thickness \delta(\rho) \rho^\beta and relaxation time function a(\rho, \chi, \chi_y) \rho^\lambda considered, where \rho \chi are density phase variable, respectively, \gamma, \alpha, \beta, \lambda\in\mathbb{R} parameters. Under some suitable assumptions on parameters \gamma , \lambda initial data, we prove global existence large-time behavior nonvacuum strong classical solutions to its Cauchy problem large data. appears be first result