Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds

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• 作者：Nguyen Thac Dung ; Nguyen Ngoc Khanh
• 关键词：Gradient estimates ; General heat equation ; Laplacian comparison theorem ; V ; Bochner–Weitzenböck ; Bakry–Emery Ricci curvature
• 刊名：Archiv der Mathematik
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：105
• 期：5
• 页码：479-490
• 全文大小：480 KB
• 参考文献：1.Brighton K.: A Liouville-type theorem for smooth metric measure spaces. Jour. Geom. Anal. 23, 562–570 (2013)MathSciNet CrossRef MATH
  2.E. Calabi, An extension of E.Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. Jour., 25 (157), 45–56
  3.Chen Q., Jost J., Qiu H. B.: Existence and Liouville theorems for V-harmonic maps from complete manifolds. Ann. Glob. Anal. Geom. 42, 565–584 (2012)MathSciNet CrossRef MATH
  4.N. T. Dung, Hamilton type gradient estimate for a nonlinear diffusion equation on smooth metric measure spaces, Manuscript
  5.Hamilton R. S.: A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1, 113–126 (1993)MathSciNet MATH
  6.G. Y. Huang and B. Q. Ma, Gradient estimates and Liouville type theorems for a nonlinear elliptic equation (2015, preprint). arXiv:​1505.​01897v1
  7.Li Y.: Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature. Nonlinear Anal. 113, 1–32 (2015)MathSciNet CrossRef MATH
  8.Li P., Yau S. T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 152–201 (1986)MathSciNet CrossRef
  9.Li X. D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. Jour. Math. Pure. Appl. 84, 1295–1361 (2005)CrossRef MATH
  10.Ruan Q. H.: Elliptic-type gradient estimates for Schrödinger equations on noncompact manifolds. Bull. London Math. Soc. 39, 982–988 (2007)CrossRef MATH
  11.Souplet P., Zhang Q. S.: Sharp grandient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38, 1045–1053 (2006)MathSciNet CrossRef MATH
  12.Wu J. Y.: Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds. Jour. Math. Anal. Appl. 369, 400–407 (2010)CrossRef MATH
  13.Wu J. Y.: Elliptic gradient estimates for a weighted heat equation and applications. Math. Zeits. 280, 451–468 (2015)CrossRef
  
• 作者单位：Nguyen Thac Dung (1)
  Nguyen Ngoc Khanh (1)
  
  1. Department of Mathematics, Mechanics and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

The purpose of this paper is to study gradient estimates of Hamilton–Souplet–Zhang type for the following general heat equation $$u_t=\Delta_V u + au\log u+bu$$on noncompact Riemannian manifolds. As its application, we show a Harnack inequality for the positive solution and a Liouville type theorem for a nonlinear elliptic equation. Our results are an extension and improvement of the work of Souplet and Zhang (Bull London Math Soc 38:1045–1053, 2006), Ruan (Bull London Math Soc 39:982–988, 2007), Li (Nonlinear Anal 113:1–32, 2015), Huang and Ma (Gradient estimates and Liouville type theorems for a nonlinear elliptic equation, Preprint, 2015), and Wu (Math Zeits 280:451–468, 2015). Keywords Gradient estimates General heat equation Laplacian comparison theorem V-Bochner–Weitzenböck Bakry–Emery Ricci curvature Mathematics Subject Classification Primary 58J35 Secondary 35B53 35K0 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (13) References1.Brighton K.: A Liouville-type theorem for smooth metric measure spaces. Jour. Geom. Anal. 23, 562–570 (2013)MathSciNetCrossRefMATH2.E. Calabi, An extension of E.Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. Jour., 25 (157), 45–563.Chen Q., Jost J., Qiu H. B.: Existence and Liouville theorems for V-harmonic maps from complete manifolds. Ann. Glob. Anal. Geom. 42, 565–584 (2012)MathSciNetCrossRefMATH4.N. T. Dung, Hamilton type gradient estimate for a nonlinear diffusion equation on smooth metric measure spaces, Manuscript5.Hamilton R. S.: A matrix Harnack estimate for the heat equation. Comm. Anal. Geom. 1, 113–126 (1993)MathSciNetMATH6.G. Y. Huang and B. Q. Ma, Gradient estimates and Liouville type theorems for a nonlinear elliptic equation (2015, preprint). arXiv:​1505.​01897v1 7.Li Y.: Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature. Nonlinear Anal. 113, 1–32 (2015)MathSciNetCrossRefMATH8.Li P., Yau S. T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 152–201 (1986)MathSciNetCrossRef9.Li X. D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. Jour. Math. Pure. Appl. 84, 1295–1361 (2005)CrossRefMATH10.Ruan Q. H.: Elliptic-type gradient estimates for Schrödinger equations on noncompact manifolds. Bull. London Math. Soc. 39, 982–988 (2007)CrossRefMATH11.Souplet P., Zhang Q. S.: Sharp grandient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38, 1045–1053 (2006)MathSciNetCrossRefMATH12.Wu J. Y.: Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds. Jour. Math. Anal. Appl. 369, 400–407 (2010)CrossRefMATH13.Wu J. Y.: Elliptic gradient estimates for a weighted heat equation and applications. Math. Zeits. 280, 451–468 (2015)CrossRef About this Article Title Gradient estimates of Hamilton–Souplet–Zhang type for a general heat equation on Riemannian manifolds Journal Archiv der Mathematik Volume 105, Issue 5 , pp 479-490 Cover Date2015-11 DOI 10.1007/s00013-015-0828-4 Print ISSN 0003-889X Online ISSN 1420-8938 Publisher Springer Basel Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Keywords Primary 58J35 Secondary 35B53 35K0 Gradient estimates General heat equation Laplacian comparison theorem V-Bochner–Weitzenböck Bakry–Emery Ricci curvature Authors Nguyen Thac Dung (1) Nguyen Ngoc Khanh (1) Author Affiliations 1. Department of Mathematics, Mechanics and Informatics (MIM), Hanoi University of Sciences (HUS-VNU), No. 334, Nguyen Trai Road, Thanh Xuan, Hanoi, Vietnam Continue reading... To view the rest of this content please follow the download PDF link above.