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.
Citation: Huashui Zhan. On anisotropic parabolic equations: from regular solutions to finite-time blow-up[J]. AIMS Mathematics, 2026, 11(3): 5299-5321. doi: 10.3934/math.2026218
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.
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Huashui Zhan. On anisotropic parabolic equations: from regular solutions to finite-time blow-up[J]. AIMS Mathematics, 2026, 11(3): 5299-5321. doi: 10.3934/math.2026218
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