@article{Zhan2026, 
author = {Huashui Zhan},
title = {On anisotropic parabolic equations: from regular solutions to finite-time blow-up},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {3},
pages = {5299-5321},
keywords = {non-Newtonian fluid equation, Poincare inequality, global solution, blow-up},
url = {https://www.sciopen.com/article/10.3934/math.2026218},
doi = {10.3934/math.2026218},
abstract = {This work established fundamental distinctions in the dynamical evolution between anisotropic and isotropic non-Newtonian fluid systems, where the harmonic mean        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                  p        ¯             was nondecreasing on              R              +      , then the local solution blowed up in finite time. The proposed methodology revealed how directional diffusivity creates distinct evolutionary patterns in solution behavior.}
}