On a non-Newtonian fluid type equation with variable diffusion coefficient

Since the non-Newtonian fluid type equations arise from a broad and in-depth background, many research achievements have been gained from 1980s. Different from the usual non-Newtonian fluid equation, there is a nonnegative variable diffusion in the equations considered in this paper. Such a variable diffusion reflects the characteristic of the medium which may not be homogenous. By giving a generalization of the Gronwall inequality, the stability and the uniqueness of weak solutions to the non-Newtonian fluid equation with variable diffusion are studied. Since the variable diffusion may be degenerate on the boundary $\mathrm{\partial }\mathrm{\Omega }$, it is found that a partial boundary value condition imposed on a submanifold of $\mathrm{\partial }\mathrm{\Omega }\times (0,T)$ is enough to ensure the well-posedness of weak solutions. The novelty is that the concept of the trace of $u(x,t)$ is generalized by a special way.