AbstractIn this paper, we consider Harnack inequalities (the gradient estimates) of positive solutions for two different heat equations via the use of the maximum principle. In the first part, we obtain the gradient estimate for positive solutions to the following nonlinear heat equation problem ∂tu=Δu+aulogu+Vu,u>0$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$ on the compact Riemannian manifold (M, g) of dimension n and with Ric(M) ≥−K. Here a > 0 and K are some constants and V is a given smooth positive function on M. Similar results are showed to be true in case when the manifold (M, g) has compact convex boundary or (M, g) is a complete non-compact Riemannian manifold. In the second part, we study Harnack inequality (gradient estimate) for positive solution to the following linear heat equation on a compact Riemannian manifold with non-negative Ricci curvature: ∂tu=Δu+∑Wiui+Vu,$ \partial _{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $ where Wi and V only depend on the space variable x ∈ M. The novelties of our paper are the refined global gradient estimates for the corresponding evolution equations, which are not previously considered by other authors such as Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010).

中文翻译：

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黎曼流形上简单热方程的 Harnack 不等式

摘要在本文中，我们通过使用最大值原理考虑了两个不同热方程的正解的 Harnack 不等式（梯度估计）。在第一部分，我们获得了以下非线性热方程问题的正解的梯度估计 ∂tu=Δu+aulogu+Vu,u>0$\partial _{t}u={\Delta } u+au\log u+Vu,~~u>0$ 在维度为 n 且 Ric(M) ≥−K 的紧致黎曼流形 (M, g) 上。这里 a > 0 和 K 是一些常数，V 是 M 上给定的平滑正函数。 如果流形 (M, g) 具有紧凑的凸边界或 (M, g) 是完整的非紧黎曼流形。在第二部分，我们在具有非负 Ricci 曲率的紧凑黎曼流形上研究以下线性热方程的正解的 Harnack 不等式（梯度估计）：∂tu=Δu+∑Wiui+Vu,$\partial_{t}u={\Delta } u+\sum W_{i}u_{i}+Vu, $ 其中 Wi 和 V 仅取决于空间变量 x ∈ M。之前被其他作者考虑过，例如 Yau (Math. Res. Lett. 2(4), 387–399, 1995), Ma (J. Funct. Anal. 241(1), 374–382, 2006), Cao et al . (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010)。我们论文的新颖之处在于对相应演化方程的精细全局梯度估计，之前其他作者如 Yau (Math. Res. Lett. 2(4), 387–399, 1995)、Ma (J . Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010)。我们论文的新颖之处在于对相应演化方程的精细全局梯度估计，之前其他作者如 Yau (Math. Res. Lett. 2(4), 387–399, 1995)、Ma (J . Funct. Anal. 241(1), 374–382, 2006), Cao et al. (J. Funct. Anal. 265, 2312–2330, 2013), Qian (Nonlinear Anal. 73, 1538–1542, 2010)。