分数阶抛物方程整体解的径向对称性与单调性

Abstract

Abstract:

This paper mainly develops the radial symmetry and monotonicity of entire solutions for fractional parabolic equations. To obtain the symmetry and monotonicity of entire solutions, the narrow region principle and maximum principle for antisymmetric functions in [9] are needed. Furthermore, to circumvent the difficulty from nonlocality for the fractional Laplacian, a fractional parabolic version of the method of moving planes will be adopted.

Key words: Fractional parabolic equations, Entire solutions, Symmetry, Monotonicity, The method of moving planes

CLC Number:

O175

Tang Yanjuan. The Radial Symmetry and Monotonicity of Entire Solutions for Fractional Parabolic Equations[J].Acta mathematica scientia,Series A, 2023, 43(5): 1409-1416.

Trendmd

References 24

[1]	Berestycki H, Nirenberg L. On the method of moving planes and the sliding method. Bol Soc Brasil Mat (NS), 1991, 22(1): 1-37
[2]	Chen W, Hu Y. Monotonicity of positive solutions for nonlocal problems in unbounded domains. J Funct Anal, 2021, 281(9): 109187 doi: 10.1016/j.jfa.2021.109187
[3]	Chen W, Li C. Classification of solutions of some nonlinear elliptic equations. Duke Math J, 1991, 63(3): 615-622
[4]	Chen W, Li C, Li Y. A direct method of moving planes for the fractional Laplacian. Adv Math, 2017, 308: 404-437 doi: 10.1016/j.aim.2016.11.038
[5]	Chen W, Li Y, Ma P. The Fractional Laplacian. Hackensack: World Scientific, 2020
[6]	Chen W, Li Y, Zhang R. A direct method of moving spheres on fractional order equations. J Funct Anal, 2017, 272(10): 4131-4157 doi: 10.1016/j.jfa.2017.02.022
[7]	Chen W, Wu L. A maximum principle on unbounded domains and a Liouville theorem for fractional $p$-harmonic functions. arXiv:1905.09986
[8]	Chen W, Wang P, Niu Y, Hu Y. Asymptotic method of moving planes for fractional parabolic equations. Adv Math, 2021, 377: 107463 doi: 10.1016/j.aim.2020.107463
[9]	Chen W, Wu L. Liouville theorems for fractional parabolic equations. Adv Nonlinear Stud, 2021, 21(4): 939-958
[10]	Du L, Tang Y. The steady incompressible jet flows issuing from a finitely long nozzle. Journal of Differential Equations, 2021, 280: 355-374
[11]	Fernández-Real X, Ros-Oton X. Regularity theory for general stable operators: Parabolic equations. J Funct Anal, 2017, 272(10): 4165-4221
[12]	Geng S, Tang Y. Convergence rates to nonlinear diffusive waves for solutions to nonlinear hyperbolic system. Acta Mathematica Scientia, 2019, 39B(1): 46-56
[13]	Hamel F, Nadirashvili N. Entire solutions of the KPP equation. Comm Pure Appl Math, 1999, 52(10): 1255-1276
[14]	Hamel F, Nadirashvili N. Travelling fronts and entire solutions of the Fisher-KPP equation in $R^{N}$. Arch Ration Mech Anal, 2001, 157(2): 91-163
[15]	Li C. Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains. Commun Partial Differ Equ, 1991, 16: 491-526
[16]	Li C. Some Qualitative Properties of Fully Nonlinear Elliptic and Parabolic Equations. New York: New York University, 1989
[17]	Li Y, Zhu M. Uniqueness theorems through the method of moving spheres. Duke Math J, 1995, 80(2): 383-417
[18]	Liu Z, Chen W. Maximum principles and monotonicity of solutions for fractional $p$-equations in unbounded domains. arXiv: 1905.06493
[19]	Pol$\acute{\rm a}\check{\rm c}$ik P. Estimates of solutions and asymptotic symmetry for parabolic equations on unbounded domains. Arch Ration Mech Anal, 2007, 183: 59-91
[20]	Pol$\acute{\rm a}\check{\rm c}$ik P. Symmetry properties of positive solutions of parabolic equations on $R^{N},$ I: Asymptotic symmetry for the Cauchy problem. Commun Partial Differ Equ, 2005, 30: 1567-1593
[21]	武乐云, 陈文雄. 具 De Giorgi 型非线性分数阶方程解的单调性. 中国科学: 数学, 2022, 52(1): 1-22
	Wu L Y, Chen W X. Monotonicity of solutions for fractional equations with De Giorgi type nonlinearities. Sci Sin Math, 2022, 52(1): 1-22
[22]	Wu L Y, Chen W X. The sliding methods for the fractional $p$-Laplacian. Adv Math, 2020, 361: Article ID 106933
[23]	Wu L Y, Chen W X. Ancient solutions to nonlocal parabolic equations. Advances in Mathematics, 2022, 408: 108607
[24]	Zhang B, Lü Z. Symmetry and nonexistence of solutions for a fully nonlinear nonlocal system. Pac J Math, 2019, 299: 237-255

The Radial Symmetry and Monotonicity of Entire Solutions for Fractional Parabolic Equations

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 24

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Chunlei He,Zihui Liu. Symmetries and Global Solutions for a Class of Hyperbolic Mean Curvature Flow [J]. Acta mathematica scientia,Series A, 2022, 42(4): 1089-1102.
[2]	Liling Lai,Jinjin Liang,Yong Chen. Complex Symmetry for a Class of Truncated Hankel Operators [J]. Acta mathematica scientia,Series A, 2022, 42(4): 961-968.
[3]	Zhenjie Li,Lei Li. A Symmetry Result for Solutions of the Fractional Laplacian with Convex Nonlinearites [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1224-1234.
[4]	Miaokun Wang,Yuming Chu,Songliang Qiu. The Simple Proof and Generalization of a Conjecture Concerning Generalized Legendre Identity [J]. Acta mathematica scientia,Series A, 2020, 40(3): 662-666.
[5]	Fang Fang,Beibei Hu,Ling Zhang. Nonlinear Integrable Couplings and Bargmann Symmetry Constraint of Super Generalized-Burgers Hierarchy [J]. Acta mathematica scientia,Series A, 2020, 40(3): 694-704.
[6]	Hongli Qian,Xiaotao Huang. A Symmetry Result for a Class of p-Laplace Involving Baouendi-Grushin Operators via Constrained Minimization Method [J]. Acta mathematica scientia,Series A, 2020, 40(3): 725-734.
[7]	Lumin Geng,Huizhan Chen,Na Li,Jipeng Cheng. The Squared Eigenfunction Symmetries and Miura Transformations for the KP and mKP Hierarchies [J]. Acta mathematica scientia,Series A, 2020, 40(1): 10-19.
[8]	Yin Li, Huang Liguo. Two Inequalities Related to Generalized Elliptic Integrals [J]. Acta mathematica scientia,Series A, 2018, 38(1): 46-53.
[9]	Zhang Ning, Xia Tiecheng. A New Integrable Nonlinear Lattice Equation Hierarchy and Their Integrable Symplectic Map [J]. Acta mathematica scientia,Series A, 2017, 37(5): 814-824.
[10]	Wang Shengjun, Han Yazhou. Generalized Picone's Identity and Its Applications for Heisenberg-Greiner Operators [J]. Acta mathematica scientia,Series A, 2017, 37(5): 895-901.
[11]	Lu Pingping, Hu Lianggen. A Note to Monotonicity Formula for Stable Solutions of the Weighted Elliptic System [J]. Acta mathematica scientia,Series A, 2017, 37(4): 698-705.
[12]	Liu Jingxin, Wu Hongxia, Zeng Yunbo. On the Ingegrable Coupled Generalization of Dispersionless BKP Hierarchy and Its Solutions [J]. Acta mathematica scientia,Series A, 2017, 37(2): 313-325.
[13]	Hu Lianggen. Liouville Type Theorems of Solutions for the Nonlinear Hénon Equations [J]. Acta mathematica scientia,Series A, 2016, 36(4): 639-648.
[14]	Zhang Pingping, Li Lin. Iteration of Upper Semi-Continuous Multifunctions on Interval [J]. Acta mathematica scientia,Series A, 2016, 36(2): 231-243.
[15]	Di Huafei, Shang Yadong. Global Existence and Nonexistence of Solutions for A Class of Fourth Order Wave Equation with Nonlinear Damping and Source Terms [J]. Acta mathematica scientia,Series A, 2015, 35(3): 618-633.