AIMS Mathematics, 2019, 4(3): 497-505. doi: 10.3934/math.2019.3.497.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Hamilton’s gradient estimate for fast diffusion equations under geometric flow

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Suppose that M is a complete noncompact Riemannian manifold of dimension n. In the present paper, we obtain a Hamilton's gradient estimate for positive solutions of the fast diffusion equations\[ \dfrac{\partial u}{\partial t}=\Delta u^m ,\qquad 1-\dfrac{4}{n+8}<m<1\]on $M\times (-\infty ,0]$ under the geometric flow.
  Figure/Table
  Supplementary
  Article Metrics

Keywords fast diffusion equation; Ricci flow; Hamilton inequality; gradient estimates

Citation: Ghodratallah Fasihi-Ramandi. Hamilton’s gradient estimate for fast diffusion equations under geometric flow. AIMS Mathematics, 2019, 4(3): 497-505. doi: 10.3934/math.2019.3.497

References

  • 1. M. Bailesteanu, X. D. Cao, A. Pulemotov, Gradient estimates for the heat equation under the Ricci flow, J. Funct. Anal., 258 (2010), 3517–3542.    
  • 2. L. R. Evangelista, E. K. Lenzi, Fractional diffusion equations and anomalous diffusion, Cambridge University Press, 2018.
  • 3. 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), 1008–1023.    
  • 4. H. Li, H. Bai, G. Zhang, Hamiltons gradient estimates for fast diffusion equations under the Ricci flow, J. Math. Anal. Appl., 444 (2016), 1372–1379.    
  • 5. G. A. Mendes, M. S. Ribeiro, R. S. Mendes, et al. Nonlinear Kramers equation associated with nonextensive statistical mechanics, Phys. Rev. E, 91 (2015), 052106.    
  • 6. P. Li, S.-T. Yau, On the parabolic kernel of the Schrdinger operator, Acta Math., 156 (1986), 153–201.    
  • 7. S. P. Liu, Gradient estimates for solutions of the heat equation under Ricci flow, Pac. J. Math., 243 (2009), 165–180.    
  • 8. J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pac. J. Math., 253 (2011), 489–510.    
  • 9. C. Tsallis, D. J. Bukman, Anomalous diffusion in the presence of external forces: Exact timedependent solutions and their thermostatistical basis, Phys. Rev. E, 54 (1996).
  • 10. J. L. Vzquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Ser. Math. Appl., Vol. 33, Oxford University Press, Oxford, 2006.

 

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved