Symmetry for variable-order fractional parabolic equations

Mengyan Chen; Huxiao Luo; Leyun Wu

doi:10.3934/dcds.2026042

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2026, Volume 52: 1-31. Doi: 10.3934/dcds.2026042

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Symmetry for variable-order fractional parabolic equations

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
2.
School of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

^*Corresponding author: Huxiao Luo
^*Corresponding author: Huxiao Luo

Received: September 17, 2025

Revised: February 11, 2026

Early access: February 28, 2026

Published online: February 28, 2026

The third author is supported by [National Natural Science Foundation of China (Grant No. 12401133) and the Guangdong Basic and Applied Basic Research Foundation (2025B151502069)].

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Abstract

This work is concerned with the symmetry of solutions to variable-order parabolic equations. We first develop several fundamental analytical tools, including maximum principles for antisymmetric functions, narrow region principles, maximum principles at infinity, and Hopf's lemma adapted to the variable-order setting. These ingredients are then employed to establish the radial symmetry and monotonicity of ancient positive solutions to variable-order fractional parabolic equations in the whole space. In particular, we show that the method of moving planes can be initiated from both $ +\infty $ and $ -\infty $ and advanced to their respective limiting positions $ \lambda_0^- $ and $ \lambda_0^+ $. By constructing suitable auxiliary functions, we further prove that $ \lambda_0^- = \lambda_0^+ $. The results apply in particular to entire solutions, namely those defined for all $ t \in \mathbb{R} $.

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Mathematics Subject Classification: Primary: 35R11, 30C80; Secondary: 35K58.

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References

[1]	H. Antil and C. N. Rautenberg, Sobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications, SIAM J. Math. Anal., 51 (2019), 2479-2503. doi: 10.1137/18M1224970.
[2]	B. Baeumer and M. M. Meerschaert, Tempered stable Lévy motion and transient super-diffusion, J. Comput. Appl. Math., 233 (2010), 2438-2448. doi: 10.1016/j.cam.2009.10.027.
[3]	B. Barrios, I. Peral, F. Soria and E. Valdinoci, A Widder's type theorem for the heat equation with nonlocal diffusion, Arch. Ration. Mech. Anal., 213 (2014), 629-650. doi: 10.1007/s00205-014-0733-1.
[4]	C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Switzerland: Springer International Publishing, 2016.
[5]	L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[6]	W. Chen, W. Dai and Y. Guo, Some recent developments on fractional parabolic equations, Discrete Contin. Dyn. Syst., 44 (2024), 2713-2791. doi: 10.3934/dcds.2024044.
[7]	W. Chen, C. Li and Y. Li, A drirect method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.
[8]	W. Chen, P. Wang, Y. Niu and Y. Hu, Asymptotic method of moving planes for fractional parabolic equations, Adv. Math., 377 (2021), 107463. doi: 10.1016/j.aim.2020.107463.
[9]	W. Chen and L. Wu, Liouville theorems for fractional parabolic equations, Adv. Nonlinear Stud., 21 (2021), 939-958. doi: 10.1515/ans-2021-2148.
[10]	W. Chen, L. Wu and P. Wang, Nonexistence of solutions for indefinite fractional parabolic equations, Adv. Math., 392 (2021), 108018. doi: 10.1016/j.aim.2021.108018.
[11]	M. D'Elia and C. Glusa, A fractional model for anomalous diffusion with increased variability: Analysis, algorithms and applications to interface problems, Numer. Methods Partial Differential Equations, 38 (2022), 2084-2103. doi: 10.1002/num.22865.
[12]	S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania), 68 (2013), 201-216.
[13]	B. Dubrulle and J.-P. Laval, Truncated Lévy laws and 2D turbulence, Eur. Phys. J. B, 4 (1998), 143-146. doi: 10.1007/s100510050362.
[14]	K. D. Dwivedi, Ra jeev, S. Das and J. F. Gomez-Aguilar, Finite difference/collocation method to solve multiterm variable-order fractional reaction-advection-diffusion equation in heterogeneous medium, Numer. Methods Partial Differential Equations, 37 (2021), 2031-2045. doi: 10.1002/num.22648.
[15]	M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3.
[16]	R. Garrappa, A. Giusti and F. Mainardi, Variable-order fractional calculus: A change of perspective, Commun. Nonlinear Sci. Numer. Simul., 102 (2021), 105904. doi: 10.1016/j.cnsns.2021.105904.
[17]	S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 2581-2615. doi: 10.3934/dcds.2014.34.2581.
[18]	M. Javanainen, H. Hammarön, L. Monticelli, J. Jeon, M. S. Miettinen, H. Martínez-Seara, R. Metzler and I. Vattulainen, Anomalous and normal diffusion of proteins and lipids in crowded lipid membranes, Faraday Discuss, 161 (2013), 397-417. doi: 10.1039/C2FD20085F.
[19]	E. K. Lenzi, H. V. Ribeiro, A. A. Tateishi, R. S. Zola and L. R. Evangelista, Anomalous diffusion and transport in heterogeneous systems separated by a membrane, Proc. R. Soc. A Math. Phys. Eng. Sci., 472 (2016), 20160502. doi: 10.1098/rspa.2016.0502.
[20]	C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29 (2002), 57-98. doi: 10.1023/A:1016586905654.
[21]	M. M. Meerschaert, Y. Zhang and B. Baeumer, Tempered anomalous diffusion in heterogeneous systems, Geophys. Res. Lett., 35 (2008), L17403. doi: 10.1029/2008GL034899.
[22]	G. Pang, P. Perdikaris, W. Cai and G. E. Karniadakis, Discovering variable fractional orders ofadvection-dispersion equations from field data using multi-fidelity Bayesian optimization, J. Comput. Phys., 348 (2017), 694-714. doi: 10.1016/j.jcp.2017.07.052.
[23]	P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\mathbb R^N$: I. Asymptotic symmetry for the Cauchy problem, Comm. Partial Differential Equations, 30 (2005), 1567-1593. doi: 10.1080/03605300500299919.
[24]	P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\mathbb{R}^N$: II. Entire solutions, Comm. Partial Differential Equations, 31 (2006), 1615-1638. doi: 10.1080/03605300600635020.
[25]	P. Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal., 183 (2007), 59-91. doi: 10.1007/s00205-006-0004-x.
[26]	S. Samko, Fractional integration and differentiation of variable order: An overview, Nonlinear Dynam., 71 (2013), 653-662. doi: 10.1007/s11071-012-0485-0.
[27]	C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer Science and Business Media, 2007.
[28]	H. Sun, A. Chang, Y. Zhang and W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2019), 27-59. doi: 10.1515/fca-2019-0003.
[29]	L. Wu and W. Chen, Ancient solutions to nonlocal parabolic equations, Adv. Math., 408 (2022), 108607. doi: 10.1016/j.aim.2022.108607.