We discuss local gradient estimates of Li and Yau type on the smooth bounded positive solutions \(w: \mathcal {M} \times [0,\infty ) \rightarrow \mathbb {R}\) to a nonlinear evolution equation \(w_t=\Delta w+a(w-w^3)\), where \(a>0\) is a constant, on a complete Riemannian manifold \(\mathcal {M}\). Global estimates are obtained from the local ones, the consequence of which will eventually yield classical Harnack inequalities for Parabolic Allen–Cahn equation and a Liouville type result for steady state solutions under the hypothesis of nonnegative Ricci curvature tensor.

中文翻译：

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Allen-Cahn 型非线性演化方程的微分 Harnack 估计

我们讨论了在平滑有界正解\(w: \mathcal {M} \times [0,\infty ) \rightarrow \mathbb {R}\)到非线性演化方程\(w_t =\Delta w+a(ww^3)\)，其中\(a>0\)是一个常数，在一个完整的黎曼流形\(\mathcal {M}\) 上。全局估计是从局部估计获得的，其结果最终将产生抛物线 Allen-Cahn 方程的经典 Harnack 不等式和非负 Ricci 曲率张量假设下稳态解的 Liouville 型结果。