In this paper, we consider the gradient estimates for a postive solution of the nonlinear parabolic equation ∂_tu = Δ_tu + hu^p on a Riemannian manifold whose metrics evolve under the geometric flow ∂_tg(t) = − 2S_g(t). To obtain these estimate, we introduce a quantity \(\underline {\boldsymbol {S}}\) along the flow which measures whether the tensor S_ij satisfies the second contracted Bianchi identity. Under conditions on Ric_g(t),S_g(t), and \(\underline {\boldsymbol {S}}\), we obtain the gradient estimates.

中文翻译：

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几何流下热型方程的Li-Yau Harnack估计

在本文中，我们考虑了非线性抛物线方程的正解的梯度估计∂_吨Ù =Δ_吨Ü + ħ Ü ^p上的黎曼流形，其度量的几何流动下演变∂_吨克（吨）= - 2小号_克（t）。为了获得这些估计，我们在流中引入了一个量\（\下划线{\ boldsymbol {S}} \），该量测量张量S _{i j}是否满足第二个压缩的Bianchi身份。在Ric _g（t），S _g（t）和\（\下划线{\ boldsymbol {S}} \），我们获得梯度估计。