본문 바로가기
KERD
논문KCI 등재

Harnack estimates for nonlinear backward heat equations with potentials along the Ricci-Bourguignon flow

Jian-Hong Wang
대한수학회 (The Korean Mathematical Society)|2020.03|대한수학회지 (Journal of the KMS)|제57권 제2호|pp. 313-329|DOI 10.4134/JKMS.j190049

0

조회

0

다운로드

0

인용

0

참고문헌

키워드

Ricci-Bourguignon flowblow upHarnack estimatenonlinear backward heat type equation
주제분류수학

초록

In this paper, we derive various differential Harnack estimates for positive solutions to the nonlinear backward heat type equations on closed manifolds coupled with the Ricci-Bourguignon flow, which was done for the Ricci flow by J.-Y. Wu \cite{[WJY12]}. The proof follows exactly the one given by X.-D. Cao \cite{[CXD08]} for the linear backward heat type equations coupled with the Ricci flow.

Abstract

In this paper, we derive various differential Harnack estimates for positive solutions to the nonlinear backward heat type equations on closed manifolds coupled with the Ricci-Bourguignon flow, which was done for the Ricci flow by J.-Y. Wu \cite{[WJY12]}. The proof follows exactly the one given by X.-D. Cao \cite{[CXD08]} for the linear backward heat type equations coupled with the Ricci flow.

참고문헌 (36)

  1. B. Andrews, "Harnack inequalities for evolving hypersurfaces", Math. Z, 217(2), pp.179~197 (1994)[DOI]
  2. J.-P. Bourguignon, "Ricci curvature and Einstein metrics, in Global dierential geometry and global analysis", Lecture Notes in Math, 838 (1981), Springer
  3. H. -D. Cao, "On Harnack's inequalities for the Kahler-Ricci ow", Invent. Math, 109(2), pp.247~263 (1992)[DOI]
  4. X. Cao, "Dierential Harnack estimates for backward heat equations with potentials under the Ricci ow", J. Funct. Anal, 255(4), pp.1024~1038 (2008)[DOI]
  5. X. Cao, "Harnack estimates for conjugate heat kernel on evolving manifolds", Math. Z, 281(1-2), pp.201~214 (2015)[DOI]
  6. X. Cao, "Dierential Harnack estimates for time-dependent heat equations with potentials", Geom. Funct. Anal, 19(4), pp.989~1000 (2009)[DOI]
  7. X. Cao, "Dierential Harnack estimates for parabolic equations, in Complex and dierential geometry", Springer Proc. Math., 8, pp.87~98 (2011), Springer
  8. G. Catino, "The Ricci-Bourguignon ow", Pacic J. Math, 287(2), pp.337~370 (2017)[DOI]
  9. H. -B. Cheng, "A new Li-Yau-Hamilton estimate for the Ricci ow", Comm. Anal. Geom., 14(3), pp.551~564 (2006)
  10. B. Chow, "The Ricci ow on the 2-sphere", J. Dierential Geom, 33(2), pp.325~334 (1991)
  11. B. Chow, "On Harnack's inequality and entropy for the Gaussian curvature ow", Comm. Pure Appl. Math, 44(4), pp.469~483 (1991)[DOI]
  12. B. Chow, "The Yamabe ow on locally conformally at manifolds with positive Ricci curvature", Comm. Pure Appl. Math, 45(8), pp.1003~1014 (1992)[DOI]
  13. B. Chow, "Constrained and linear Harnack inequalities for parabolic equations", Invent. Math, 129(2), pp.213~238 (1997)[DOI]
  14. B. Chow, "New Li-Yau-Hamilton inequalities for the Ricci ow via the space-time approach", J. Dierential Geom, 60(1), pp.1~54 (2002)
  15. A. E. Fischer, "An introduction to conformal Ricci ow", Classical Quantum Gravity, 21(3), pp.S171~S218 (2004)[DOI]
  16. C. M. Guenther, "The fundamental solution on manifolds with time-dependent metrics", J. Geom. Anal, 12(3), pp.425~436 (2002)[DOI]
  17. H. Guo, "Harnack estimates for nonlinear backward heat equations in geometric ows", J. Funct. Anal, 267(8), pp.2638~2662 (2014)[DOI]
  18. H. Guo, "Harnack estimates for nonlinear heat equations with potentials in geometric ows", Manuscripta Math, 148(3-4), pp.471~484 (2015)[DOI]
  19. R. S. Hamilton, "The Ricci ow on surfaces, in Mathematics and general relativity", Contemp. Math, 71 (1988)[DOI]
  20. R. S. Hamilton, "The Harnack estimate for the Ricci ow", J. Dierential Geom, 37(1), pp.225~243 (1993)
  21. R. S. Hamilton, "A matrix Harnack estimate for the heat equation", Comm. Anal. Geom, 1(1), pp.113~126 (1993)[DOI]
  22. R. S. Hamilton, "Harnack estimate for the mean curvature ow", J. Dierential Geom, 41(1), pp.215~226 (1995)
  23. S. Kuang, "A gradient estimate for all positive solutions of the conjugate heat equation under Ricci ow", J. Funct. Anal, 255(4), pp.1008~1023 (2008)[DOI]
  24. P. Li, "On the parabolic kernel of the Schrodinger operator", Acta Math, 156(3-4), pp.153~201 (1986)
  25. L. Ni, "A note on Perelman's LYH-type inequality", Comm. Anal. Geom, 14(5), pp.883~905 (2006)
  26. L. Ni, "A matrix Li-Yau-Hamilton estimate for Kahler-Ricci ow", J. Dierential Geom, 75(2), pp.303~358 (2007)
  27. G. Perelman, "The entropy formula for the Ricci ow and its geometric applications", arXiv math.DG, 2002
  28. N. Sesum, "Notes on Perelman's paper on the entropy formula for the Ricci ow and its geometric applications", ~ (2003)
  29. J. -Y. Wu, "Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds", J. Math. Anal. Appl, 369(1), pp.400~407 (2010)[DOI]
  30. J. -Y. Wu, "Dierential Harnack inequalities for nonlinear heat equations with potentials under the Ricci ow", Pacic J. Math, 257(1), pp.199~218 (2012)[DOI]
  31. J. -Y. Wu, "Interpolating between constrained Li-Yau and Chow-Hamilton Harnack inequalities on a surface", Arch. Math. (Basel), 94(6), pp.591~600 (2010)[DOI]
  32. Y. Yang, "Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds", Proc. Amer. Math. Soc, 136(11), pp.4095~4102 (2008)[DOI]
  33. S. -T. Yau, "On the Harnack inequalities of partial dierential equations", Comm. Anal. Geom, 2(3), pp.431~450 (1994)[DOI]
  34. S. -T. Yau, "Harnack inequality for non-self-adjoint evolution equations", Math. Res. Lett, 2(4), pp.387~399 (1995)[DOI]
  35. R. Ye, "Global existence and convergence of Yamabe ow", J. Dierential Geom, 39(1), pp.35~50 (1994)
  36. Q. S. Zhang, "Some gradient estimates for the heat equation on domains and for an equation by Perelman", Int. Math. Res. Not, 2006, pp.39~ (2006)[DOI]

관련 논문AI추천

Harnack Estimate for Positive Solutions to a Nonlinear Equation Under Geometric FlowGhodratallah Fasihi-Ramandi, Shahroud Azami경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2021 · 제61권 제3호An interpolating Harnack inequality for nonlinear heat equation on a surfaceChengzhe Zhu, Hongxin Guo대한수학회 · 대한수학회보2021 · 제58권 제4호Harnack inequality for parabolic Yamabe-type equation under integral curvature condition on manifoldsLiang Zhao대한수학회 · 대한수학회보2025 · 제62권 제1호Li-Yau gradient estimates on closed manifolds under Bakry-\'Emery Ricci curvature conditionsLing Wu, XingYu Song대한수학회 · 대한수학회지2023 · 제60권 제5호Gradient estimates and Harnack inequalites of nonlinear heat equations for the $V$-LaplacianHa Tuan Dung대한수학회 · 대한수학회지2018 · 제55권 제6호Evolution and monotonicity for a class of quantities along the Ricci-Bourguignon flowAsadollah Razavi, Farzad Daneshvar대한수학회 · 대한수학회지2019 · 제56권 제6호

이 간행물의 다른 논문 (13건)

Modular Jordan type for $\k [x, y] / (x^m,y^n)$ for $m = 3, 4$신용수, 박정필pp. 283-312Applications of differential subordinations to certain classes of starlike functionsShagun Banga, S. SIVAPRASAD KUMARpp. 331-357Conditional expectation of Pettis integrable unbounded random setsMohamed El Haramipp. 359-381Orbit equivalence on self-similar groups and their $C^*$-algebras이인협pp. 383-399An efficient algorithm for sliding window based incremental principal components analysis이근섭pp. 401-414
원문 보기