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The existence and uniqueness of weak solutions are studied to the initial Dirichlet problem of the equationwith inf p(x) > 2. The problems describe the motion of generalized Newtonian fluids which were studied by some other authors in which the exponent p was required to satisfy a logarithmic Hölder continuity condition. The authors in this paper use a difference scheme to transform the parabolic problem to a sequence of elliptic problems and then obtain the existence of solutions with less constraint to p(x). The uniqueness is also proved.

The aims of this paper are to discuss existence and uniqueness of local solutions for a class of nonNewtonian fluids with singularity and vacuum in one-dimensional bounded intervals. There are two important points in this paper, one is that we allow the initial vacuum; another one is that the viscosity term of momentum equation is with singularity and fully nonlinearity.

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