대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제50권5호
  • /
  • Pages.1587-1598
  • /
  • 2013
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

HARNACK INEQUALITY FOR A NONLINEAR PARABOLIC EQUATION UNDER GEOMETRIC FLOW

  • Zhao, Liang (Department of Mathematics Nanjing University of Aeronautics and Astronautics)
  • 투고 : 2012.08.27
  • 발행 : 2013.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation $$\frac{{\partial}u}{{\partial}t}={\triangle}u-b(x,t)u^{\sigma}$$ under general geometric flow on complete noncompact manifolds, where 0 < ${\sigma}$ < 1 is a real constant and $b(x,t)$ is a function which is $C^2$ in the $x$-variable and $C^1$ in the$t$-variable. As an application, we get an interesting Harnack inequality.

키워드

참고문헌

  1. T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampre equations, Springer, New York, 1982.
  2. E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), no. 1, 45-56. https://doi.org/10.1215/S0012-7094-58-02505-5
  3. R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom. 1 (1993), no. 1, 113-126. https://doi.org/10.4310/CAG.1993.v1.n1.a6
  4. S. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal. 255 (2008), no. 4, 1008-1023. https://doi.org/10.1016/j.jfa.2008.05.014
  5. P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), no. 3-4, 153-201. https://doi.org/10.1007/BF02399203
  6. S. P. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pacific J. Math. 243 (2009), no. 1, 165-180. https://doi.org/10.2140/pjm.2009.243.165
  7. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint, (2002), arXiv math.DG/0211159.
  8. J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math. 253 (2011), no. 2, 489-510. https://doi.org/10.2140/pjm.2011.253.489
  9. J. Zhang and B. Q. Ma, Gradient estimates for a nonlinear equation ${\Delta}_fu+cu^{-{\alpha}}$= 0 on complete noncompact manifolds, Commun. Math. 19 (2011), no. 1, 73-84.
  10. X. B. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Anal. 74 (2011), no. 15, 5141-5146. https://doi.org/10.1016/j.na.2011.05.008