On anisotropic parabolic equations: from regular solutions to finite-time blow-up

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.