The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space

Yubo Ni

doi:doi:10.11648/j.ajam.20251301.12

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The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space

Yubo Ni^*

Published in American Journal of Applied Mathematics (Volume 13, Issue 1)

Received: 18 December 2024 Accepted: 30 December 2024 Published: 14 January 2025

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Abstract

This paper is mainly concerned with the symmetry and monotonicity of solutions to a fractional parabolic Kirchhoff equation. We first establishes the asymptotic narrow region principle, the asymptotic maximum principle near infinity, the asymptotic strong maximum principle and the Hopf principle for antisymmetric functions in bounded and unbounded domains. By the method of moving plane, it then derives the symmetry of positive solutions on the unit sphere and in the entire space. Next, we point out how to apply these tools and methods to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. By some researches, we find that no matter how we set the initial value, it will not affect the property of the solution approaching a radially symmetric function as t approaches infinity. Throughout the paper, establishing the maximum principle plays a central role in exploring and studying the fractional parabolic Kirchhoff equation. After establishing different maximum principles, one can study the properties of a solution to the parabolic equation under different conditions. Finally, the novelty of this article is that it is the first time to apply method of moving plane to fractional parabolic Kirchhoff problems and the ideas and methods presented in this article are applicable to studying different non local parabolic problems, various operators and the symmetry of solutions in different regions.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 1)
DOI	10.11648/j.ajam.20251301.12
Page(s)	13-29
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Fractional Laplace Operator, Maximum Principle, Moving Plane Method, Parabolic Equation

References

[1]	KIRCHHOFF G. Mechanik [M]. Teubner, Leipzig, 1883.
[2]	POHOZAEV S I. A certain class of quasilinear hyperbolic equations [J]. Mat. Sb. (N,s.), 1975, 96(138): 152-166. https://doi.org/10.1070/SM1975V025N01ABEH002203
[3]	LIONS JL. On some questions in boundary value problems of mathematical physics [J]. NorthHolland Math. Stud, 1978, 30: 284-346. https://doi.org/10.1016/S0304-0208(08)70870-3
[4]	ALVES C O, CORREA F. J. S. A. On existence of solutions for a class of problem involving a nonlinear operator [J]. Commun. Appl. Nonlinear Anal, 2001, 8: 43-56. https://doi.org/10.1007/s13540-024-00296-y
[5]	ANCONA P D, SPAGNOLO S. Global solvability for the degenerate Kirchhoff equation with real analytic data [J]. Invent. Math, 1992, 108: 247-262. https://doi.org/10.1007/BF02100605
[6]	LI G, NIU Y. The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation [J]. Acta Math. Sci, 2020, 40B(1): 1-23. https://doi.org/10.1007/s10473-020-0107-y
[7]	PUCCI P, SALDI S. Critical stationary Kirchhoff equations in Rn involving nonlocal operators [J]. Rev. Mat. Iberoam, 2016, 32: 1-22. https://doi.org/10.4171/rmi/879
[8]	HE X, ZOU W. Ground state solutions for a class of fractional Kirchhoff equations with critical growth [J]. Sci. China Math, 2019, 62(5): 853-890. https://doi.org/10.1007/s11425-017-9399-6
[9]	ZHANG B, WANG L. Existence results for Kirchhoff- type superlinear problems involving the fractional Laplacian [J]. Proc. Roy. Soc. Edinburgh Sect. A, 2019, 149: 1061-1081. https://doi.org/10.1017/prm.2018.105
[10]	CAFFARELLI L, SILVESTRE L. Regularity theory for fully nonlinear integro-differential equations [J]. Comm, Pure Appl. Math, 2009, 62(5): 597-638. https://doi.org/10.1002/cpa.20274
[11]	CHEN W, LI C, OU B. Classification of solutions for an integral equation [J]. Comm, Pure Appl. Math, 2006, 59(3): 330-343. https://doi.org/10.1002/cpa.20116
[12]	JAROHS S, WETH T. Symmetry via antisymmetric maximum principles in nonlocal problems of variable order [J]. Annali di Matematica Pura ed Applicate, 2016, 195(1): 273-291. https://doi.org/10.1007/s10231-014-0462-y
[13]	CHEN W, LI C. Maximum principle for the fractional p-Laplacian and symmetry of solutions [J]. Adv. Math, 2018, 335: 735-758. https://doi.org/10.1016/J.AIM.2018.07.016
[14]	CHEN W, LI C, LI Y. A direct method of moving planes for the fractional Laplacian [J]. Adv. Math, 2017, 308: 404-437. https://doi.org/10.1016/j.aim.2016.11.038
[15]	CHEN W, WANG P, NIU Y, et al. Asymptotic method of moving planes for fractional parabolic equations [J]. Adv. Math, 2021, 377(1). https://doi.org/10.1016/j.aim.2020.107463
[16]	NIU Y. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations [J]. Commun. Pure Appl. Anal, 2021, 20(4): 1431-1445. https://doi.org/10.3934/CPAA.2021027
[17]	FERNÁNDEZ-REAL X, ROS-OTON X. Regularity theory for general stable operator: parabolic equation [J]. J. Funct. Anal, 2017, 272(10): 4165-4221. https://doi.org/10.1016/J.JFA.2017.02.015
[18]	WANG P, CHEN W. Hopf’s lemmas for parabolic fractional Laplacians and parabolic fractional p- Laplacians [J]. Commun. Pure Appl. Anal, 2022, 21(9): 3055-3069. https://doi.org/10.48550/arXiv.2010.01212

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APA Style

Ni, Y. (2025). The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. American Journal of Applied Mathematics, 13(1), 13-29. https://doi.org/10.11648/j.ajam.20251301.12

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ACS Style

Ni, Y. The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. Am. J. Appl. Math. 2025, 13(1), 13-29. doi: 10.11648/j.ajam.20251301.12

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AMA Style

Ni Y. The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. Am J Appl Math. 2025;13(1):13-29. doi: 10.11648/j.ajam.20251301.12

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@article{10.11648/j.ajam.20251301.12,
  author = {Yubo Ni},
  title = {The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {1},
  pages = {13-29},
  doi = {10.11648/j.ajam.20251301.12},
  url = {https://doi.org/10.11648/j.ajam.20251301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251301.12},
  abstract = {This paper is mainly concerned with the symmetry and monotonicity of solutions to a fractional parabolic Kirchhoff equation. We first establishes the asymptotic narrow region principle, the asymptotic maximum principle near infinity, the asymptotic strong maximum principle and the Hopf principle for antisymmetric functions in bounded and unbounded domains. By the method of moving plane, it then derives the symmetry of positive solutions on the unit sphere and in the entire space. Next, we point out how to apply these tools and methods to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. By some researches, we find that no matter how we set the initial value, it will not affect the property of the solution approaching a radially symmetric function as t approaches infinity. Throughout the paper, establishing the maximum principle plays a central role in exploring and studying the fractional parabolic Kirchhoff equation. After establishing different maximum principles, one can study the properties of a solution to the parabolic equation under different conditions. Finally, the novelty of this article is that it is the first time to apply method of moving plane to fractional parabolic Kirchhoff problems and the ideas and methods presented in this article are applicable to studying different non local parabolic problems, various operators and the symmetry of solutions in different regions.},
 year = {2025}
}

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TY  - JOUR
T1  - The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space
AU  - Yubo Ni
Y1  - 2025/01/14
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251301.12
DO  - 10.11648/j.ajam.20251301.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 13
EP  - 29
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251301.12
AB  - This paper is mainly concerned with the symmetry and monotonicity of solutions to a fractional parabolic Kirchhoff equation. We first establishes the asymptotic narrow region principle, the asymptotic maximum principle near infinity, the asymptotic strong maximum principle and the Hopf principle for antisymmetric functions in bounded and unbounded domains. By the method of moving plane, it then derives the symmetry of positive solutions on the unit sphere and in the entire space. Next, we point out how to apply these tools and methods to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. By some researches, we find that no matter how we set the initial value, it will not affect the property of the solution approaching a radially symmetric function as t approaches infinity. Throughout the paper, establishing the maximum principle plays a central role in exploring and studying the fractional parabolic Kirchhoff equation. After establishing different maximum principles, one can study the properties of a solution to the parabolic equation under different conditions. Finally, the novelty of this article is that it is the first time to apply method of moving plane to fractional parabolic Kirchhoff problems and the ideas and methods presented in this article are applicable to studying different non local parabolic problems, various operators and the symmetry of solutions in different regions.
VL  - 13
IS  - 1
ER  -

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Author Information

Yubo Ni

School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, China

Contact Email

http://orcid.org/0009-0008-2321-0015

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APA Style

Ni, Y. (2025). The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. American Journal of Applied Mathematics, 13(1), 13-29. https://doi.org/10.11648/j.ajam.20251301.12

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ACS Style

Ni, Y. The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. Am. J. Appl. Math. 2025, 13(1), 13-29. doi: 10.11648/j.ajam.20251301.12

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AMA Style

Ni Y. The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space. Am J Appl Math. 2025;13(1):13-29. doi: 10.11648/j.ajam.20251301.12

Copy | Download

@article{10.11648/j.ajam.20251301.12,
  author = {Yubo Ni},
  title = {The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {1},
  pages = {13-29},
  doi = {10.11648/j.ajam.20251301.12},
  url = {https://doi.org/10.11648/j.ajam.20251301.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251301.12},
  abstract = {This paper is mainly concerned with the symmetry and monotonicity of solutions to a fractional parabolic Kirchhoff equation. We first establishes the asymptotic narrow region principle, the asymptotic maximum principle near infinity, the asymptotic strong maximum principle and the Hopf principle for antisymmetric functions in bounded and unbounded domains. By the method of moving plane, it then derives the symmetry of positive solutions on the unit sphere and in the entire space. Next, we point out how to apply these tools and methods to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. By some researches, we find that no matter how we set the initial value, it will not affect the property of the solution approaching a radially symmetric function as t approaches infinity. Throughout the paper, establishing the maximum principle plays a central role in exploring and studying the fractional parabolic Kirchhoff equation. After establishing different maximum principles, one can study the properties of a solution to the parabolic equation under different conditions. Finally, the novelty of this article is that it is the first time to apply method of moving plane to fractional parabolic Kirchhoff problems and the ideas and methods presented in this article are applicable to studying different non local parabolic problems, various operators and the symmetry of solutions in different regions.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Symmetry of Solutions for a Class of KIRCHHOFF Equations on the Unit Ball and in the Entire Space
AU  - Yubo Ni
Y1  - 2025/01/14
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251301.12
DO  - 10.11648/j.ajam.20251301.12
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 13
EP  - 29
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251301.12
AB  - This paper is mainly concerned with the symmetry and monotonicity of solutions to a fractional parabolic Kirchhoff equation. We first establishes the asymptotic narrow region principle, the asymptotic maximum principle near infinity, the asymptotic strong maximum principle and the Hopf principle for antisymmetric functions in bounded and unbounded domains. By the method of moving plane, it then derives the symmetry of positive solutions on the unit sphere and in the entire space. Next, we point out how to apply these tools and methods to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. By some researches, we find that no matter how we set the initial value, it will not affect the property of the solution approaching a radially symmetric function as t approaches infinity. Throughout the paper, establishing the maximum principle plays a central role in exploring and studying the fractional parabolic Kirchhoff equation. After establishing different maximum principles, one can study the properties of a solution to the parabolic equation under different conditions. Finally, the novelty of this article is that it is the first time to apply method of moving plane to fractional parabolic Kirchhoff problems and the ideas and methods presented in this article are applicable to studying different non local parabolic problems, various operators and the symmetry of solutions in different regions.
VL  - 13
IS  - 1
ER  -

Copy | Download