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Hamilton and Souplet–Zhang type estimations on semilinear parabolic system along geometric flow

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Abstract

In this article we derive both Hamilton type and Souplet–Zhang type gradient estimations for a system of semilinear equations along a geometric flow on a weighted Riemannian manifold.

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References

  1. S. Azami, Gradient estimates for a weighted parabolic equation under geometric flow, arXiv:2112.01271v1 [math.DG], 2021.

  2. D. Bakry and M. Émery, Diffusions hypercontractives, Seminaire de probabilities XIX 1983/84, Lecture notes in Math., Springer (1123) (1985), 177-206.

  3. E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J., 25(1) (1958), 45-56.

  4. Y. -H. Feng and C. -M. Liu, Stability of steady-state solutions to Navier-Stokes-Poisson systems, Journal of Mathematical Analysis and Applications, 462(2) (2018), 1679-1694.

    Article  MathSciNet  Google Scholar 

  5. R. Hamilton, Three-manifolds with positive Ricci curvature, Journal of Differential Geometry, 17(2)(1982), 255-306.

    Article  MathSciNet  Google Scholar 

  6. R. Hamilton, The Harnack estimate for the Ricci flow, Journal of Differential Geometry, 37 (1993), 225-243.

    Article  MathSciNet  Google Scholar 

  7. S. K. Hui, A. Saha and S. Bhattacharyya, Hamilton and Souplet-Zhang type gradient estimate along geometric flow, Filomat 37 (12) (2023), 3935-3945.

    Article  MathSciNet  Google Scholar 

  8. S. K. Hui, A. Abolarinwa and S. Bhattacharyya, Gradient estimations for nonlinear elliptic equations on weighted Riemannian manifolds, Lobachevskii Journal of Mathematics, 44 (2023), 1341–1349.

    Article  MathSciNet  Google Scholar 

  9. X. D. Li, Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds, J. Math. Pure. Appl., 84(10) (2005), 1295-1361.

    Article  MathSciNet  Google Scholar 

  10. F. Li and Y. Bao, Uniform stability of the solution for a memory type elasticity system with nonhomogeneous boundary control condition, Journal of Dynamical and Control Systems, 23 (2) (2017), 301-315.

    Article  MathSciNet  Google Scholar 

  11. P. Li and S.-T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153-201.

    Article  MathSciNet  Google Scholar 

  12. R. Scohen and S.-T. Yau, Lecture on differential geometry, International Press, Cambridge, MA, 1994.

    Google Scholar 

  13. X. Shen, J. Ding, Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions, Computers and Mathematics with Applications, 77 (2019), 3250-3263.

    Article  MathSciNet  Google Scholar 

  14. P. Souplet and Q. S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc., 38 (2006), 1045-1053.

    Article  MathSciNet  Google Scholar 

  15. J. Sun, Gradient estimates for positive solutions of the heat equation under geometric flow, Pacific J. Math., 253(2) (2011), 489-510.

    Article  MathSciNet  Google Scholar 

  16. M. Winkler, Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pure. Appl., 100 (2013), 748-767.

    Article  MathSciNet  Google Scholar 

  17. H. Wu, Differential Harnack Estimates for a Semilinear Parabolic System, Hindawi Journal of Function Spaces, 2019, Article ID 1314268, https://doi.org/10.1155/2019/1314268

  18. J.-Y. Wu, Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds, J. Math. Anal. Appl., 369 (2010), 400-407.

    Article  MathSciNet  Google Scholar 

  19. H. Wu and X. Yang, Global existence and finite time blow-up for a parabolic system on hyperbolic space, Journal of Mathematical Physics, 59 (2018), 1-11.

    Article  MathSciNet  Google Scholar 

  20. X. Zheng, Y. Shang, and H. Di, The time-periodic solutions to the modified Zakharov equations with a quantum correction, Mediterranean Journal of Mathematics, 14 (2017), 1-17.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The University of Burdwan, Golapbag, Burdwan, West Bengal, 713104, India

    Sujit Bhattacharyya & Shyamal Kumar Hui

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

    Shahroud Azami

Authors
  1. Sujit Bhattacharyya
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  2. Shahroud Azami
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  3. Shyamal Kumar Hui
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Corresponding author

Correspondence to Shyamal Kumar Hui.

Additional information

Communicated by Indranil Biswas.

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Bhattacharyya, S., Azami, S. & Hui, S.K. Hamilton and Souplet–Zhang type estimations on semilinear parabolic system along geometric flow. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00586-4

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  • DOI: https://doi.org/10.1007/s13226-024-00586-4

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