Nonlinearly exponential stability for the compressible Navier-Stokes equations with temperature-dependent transport coefficients

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This paper is concerned with an initial and boundary value problem of the compressible Navier-Stokes equations for one-dimensional viscous and heat-conducting ideal polytropic fluids with temperature-dependent transport coefficients. In the case when the viscosity μ(θ)=θα and the heat-conductivity κ(θ)=θβ with α,β∈[0,∞), we prove the global-in-time existence of strong solutions under some assumptions on the growth exponent α and the initial data. As a byproduct, the nonlinearly exponential stability of the solution is obtained. It is worth pointing out that the initial data could be large if α≥0 is small, and the growth exponent β≥0 can be arbitrarily large.