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The one-dimensional compressible non-Newtonian models are considered in this paper.The extra-stress tensor in our models satisfies a kind of power law structure which was proposed by O.A.Ladyzhenskaya in 1970s.In particular,the viscosity coefficient in our models depends on the density.By using energy-estimate,the authors obtain the existence and uniqueness of local strong solutions for which the density is non-negative.