This work established fundamental distinctions in the dynamical evolution between anisotropic and isotropic non-Newtonian fluid systems, where the harmonic mean $\overline{p}$ of $\{p_i{\}}_{i=1}^N$ emerged as a critical bifurcation parameter governing solution behaviors. Using the parabolic regularization method, we established the local existence of a weak solution. By applying the Poincar inequality in the single-variable sense and imposing certain restrictions on the nonlinear term $f(x,t,u,u_{x_i})$, we proved the existence of a global solution. Moreover, if $f(x,t,u,u_{x_i})=f(u)$ and $f(u)/u^{\overline{p}}$ was nondecreasing on ${\mathbb{R}}^{+}$, then the local solution blowed up in finite time. The proposed methodology revealed how directional diffusivity creates distinct evolutionary patterns in solution behavior.

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.