Gradient estimates for a nonlinear equation under the almost Ricci soliton condition

Document Type : Research Paper

Authors

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

2 Department of pure Mathematics, Faculty of Sciences Imam Khomeini International University, Qazvin, Iran.

Abstract

In this paper, we study the gradient estimate for the positive solutions of the equation $\Delta u+au(\log u)^{p}+bu=f$ on an almost Ricci soliton $(M^{n},g,X,\lambda)$. In a special case, when $X=\nabla h$ for a smooth function $h$, we derive a gradient estimate for an almost gradient Ricci soliton.

Keywords

Main Subjects

References

  • [1] S. Azami and S. Hajiaghasi, New volume comparison with almost Ricci soliton, Communications of the Korean Mathematical Society, 37(3) (2022), 839-849.
  • [2] R. H. Bamler, Entropy and heat kernel bounds on a Ricci flow background, arXiv:2008.07093v3, (2021).
  • [3] A. Barros and E. Ribeiro JR, Some characterizations for compact almost Ricci solitons, American Mathematical Society, 140(3) (2012), 1033-1040.
  • [4] X. Cao and R. S. Hamilton, Differential Harnack estimates for time-dependent heat equations, Geom. Funct. Anal., 19(4) (2009), 989-1000.
  • [5] X. Dai, G. Wei, and Z. Zhang, Local Sobolev constant estimate for integral Ricci curvature bounds, Advances in Mathematics, 325 (2018), 1-33.
  • [6] S. Deshmukh, H. Alsodais, and N. Bin Turki, Some results on Ricci almost solitons, Symmetry, 13(3) (2021).
  • [7] S. Deshmukh and H. Alsodais, A note on almost Ricci soliton, Anal. Math. Phy., (2020).
  • [8] S. Deshmukh, Almost Ricci solitons isometric to spheres, International Journal or Geometric Methods in Modern Physics, 16(5) (2019).
  • [9] S. Hajiaghasi and S. Azami, Gradient estimate of Poisson equation under the almost Ricci solitons, Boundary Value Problems, (2022), 104.
  • [10] Q. Han and F. Li, Elliptic partial differential equation, Courant. Inst. Math. Sci.; Amer. Math. Soc., 1997.
  • [11] W. Hebisch and L. Saloff-Costa, On the relation between elliptic and parabolic Harnack inequations, Ann. Inst. Fourier, Grenoble, 51(5) (2001), 1437-1481.
  • [12] P. Li and S. T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Mathematic, (156) (1986), 153-201.
  • [13] B. Peng, Y. Wang, and G. Wei, Yau type gradient estimates for $∆u + au(logu)p + bu = 0$ on Riemannian manifolds, Journal of mathematical analysis and application, 498(1) (2021).
  • [14] S. Pigola, M. Rigoli, M. Rimoldi, and A. G. Setti, Ricci almost solitons, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5(10) (2011), 757-799.
  • [15] C. Rose, Heat kernel upper bound on Riemannian manifolds with locally uniform Ricci curvature integral bounds, J. Geom. Anal., 27(2) (2017), 1737-1750.
  • [16] R. Sharma, Almost Ricci solitons and K-contact geometry, Monatsh. Math., 175 (2014), 621-628.
  • [17] Q. S. Zhang, Some gradient estimates for the heat equation on domains and for an equation by Perelman, Int. Math. Res. Not., 39 (2006).
  • [18] Q. S. Zhang and M. Zhu, New volume comparison results and applications to degeneration of Riemannian metrics, Advances in Mathematics, 352 (2019), 1096-1154.
Computational Methods for Differential Equations
Volume 12, Issue 4
October 2024
Pages 710-718

Files

History

  • Receive Date: 10 April 2023
  • Revise Date: 22 February 2024
  • Accept Date: 27 March 2024

Share

How to cite

Statistics