Yamabe流下一类非线性抛物型方程的梯度估计
Gradient Estimates for a Nonlinear Parabolic Equation under the Yamabe Flow
DOI: 10.12677/aam.2026.154134, PDF,   
作者: 黄陈林*, 杨 飞：中国地质大学数学与物理学院，湖北 武汉
关键词: 梯度估计；抛物型方程；Harnack不等式；Gradient Estimates； Parabolic Equations； Harnack Inequality
摘要: 本文研究Yamabe曲率流下非线性抛物方程 ( Δ ϕ − ∂ t )u=au ( lnu ) α 正解的梯度估计，得到了该方程的Hamilton型、Li-Yau型和Souplet-Zhang型三种梯度估计。本文将相关结果系统性地推广至更一般的非线性抛物方程情形。
Abstract: This paper addresses the issue of the limited universality of gradient estimation methods for positive solutions to nonlinear parabolic equations. We establish gradient estimates for positive solutions of the equation ( Δ ϕ − ∂ t )u=au ( lnu ) α obtaining three types of estimates: the Hamilton-type, Li-Yau-type, and Souplet-Zhang-type. Our work systematically extends the relevant results to a broader class of nonlinear parabolic equations.
文章引用：黄陈林, 杨飞. Yamabe流下一类非线性抛物型方程的梯度估计[J]. 应用数学进展, 2026, 15(4): 42-50. https://doi.org/10.12677/aam.2026.154134

参考文献

[1] Li, P. and Yau, S.T. (1986) On the Parabolic Kernel of the Schrödinger Operator. Acta Mathematica, 156, 153-201. [Google Scholar] [CrossRef] 
[2] Hamilton, R.S. (1993) Matrix Harnack Estimate for the Heat Equation. Communications in Analysis and Geometry, 1, 113-126. [Google Scholar] [CrossRef] 
[3] Kotschwar, B. (2007) Hamilton’s Gradient Estimate for the Heat Kernel on Complete Manifolds. Proceedings of the American Mathematical Society, 135, 3013-3019. [Google Scholar] [CrossRef] 
[4] Souplet, P. and Zhang, Q.S. (2006) Sharp Gradient Estimate and Yau’s Liouville Theorem for the Heat Equation on Noncompact Manifolds. Bulletin of the London Mathematical Society, 38, 1045-1053. [Google Scholar] [CrossRef] 
[5] Bakry, D. and Qian, Z.M. (1999) Harnack Inequalities on a Manifold with Positive or Negative Ricci Curvature. Revista Matemática Iberoamericana, 15, 143-179. [Google Scholar] [CrossRef] 
[6] Petersen, P. and Wei, G. (2000) Analysis and Geometry on Manifolds with Integral Ricci Curvature Bounds. II. Transactions of the American Mathematical Society, 353, 457-478. [Google Scholar] [CrossRef] 
[7] Li, M. (2006) Gradient Estimates for a Simple Elliptic Equation on Complete Non-Compact Riemannian Manifolds. Journal of Functional Analysis, 241, 374-382. [Google Scholar] [CrossRef] 
[8] Azami, S. (2024) Differential Gradient Estimates for Nonlinear Parabolic Equations under Integral Ricci Curvature Bounds. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 118, Article No. 51. [Google Scholar] [CrossRef] 
[9] Li, Y. and Zhu, X. (2016) Harnack Estimates for a Heat-Type Equation under the Ricci Flow. Journal of Differential Equations, 260, 3270-3301. [Google Scholar] [CrossRef] 
[10] Zhang, L. (2020) Local Parabolic and Elliptic Gradient Estimates for a Generalized Heat-Type Equation under the Yamabe Flow. Journal of Mathematical Analysis and Applications, 485, Article ID: 123770. [Google Scholar] [CrossRef] 
[11] Li, J.Y. (1991) Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic and Nonlinear Elliptic Equations on Riemannian Manifolds. Journal of Functional Analysis, 100, 233-256. [Google Scholar] [CrossRef] 
[12] Wu, J.Y. (2017) Elliptic Gradient Estimates for a Nonlinear Heat Equation and Applications. Nonlinear Analysis: Theory, Methods & Applications, 151, 1-17. [Google Scholar] [CrossRef] 
[13] Bakry, D. and Ledoux, M. (1996) Sobolev Inequalities and Myers’s Diameter Theorem for an Abstract Markov Generator. Duke Mathematical Journal, 85, 252-270. [Google Scholar] [CrossRef] 
[14] Qian, Z. (1997) Estimates for Weighted Volumes and Applications. The Quarterly Journal of Mathematics, 48, 235-242. [Google Scholar] [CrossRef] 
[15] Yang, F. and Zhang, L. (2019) Gradient Estimates for a Nonlinear Parabolic Equation on Smooth Metric Measure Spaces. Nonlinear Analysis, 187, 49-70. [Google Scholar] [CrossRef] 
[16] Chen, L. and Chen, W. (2008) Gradient Estimates for a Nonlinear Parabolic Equation on Complete Non-Compact Riemannian Manifolds. Annals of Global Analysis and Geometry, 35, 397-404. [Google Scholar] [CrossRef] 
[17] Yau, S.T. (1975) Harmonic Functions on Complete Riemannian Manifolds. Communications on Pure and Applied Mathematics, 28, 201-228. [Google Scholar] [CrossRef] 
[18] Petersen, P. and Wei, G.F. (1997) Relative Volume Comparison with Integral Curvature Bounds. Geometric and Functional Analysis, 7, 1031-1045. [Google Scholar] [CrossRef] 

为你推荐