时空分数阶波方程解的渐近性与长时间行为

doi:10.12066/j.issn.1007-2861.2257

上海大学学报(自然科学版) ›› 2021, Vol. 27 ›› Issue (6): 1149-1161.doi: 10.12066/j.issn.1007-2861.2257

时空分数阶波方程解的渐近性与长时间行为

李志强¹^,²()

1.上海大学理学院, 上海 200444
2.吕梁学院数学系, 山西吕梁 033001

收稿日期:2020-05-08 出版日期:2021-12-31 发布日期:2020-09-23
通讯作者: 李志强 E-mail:lizhiqiang0914@126.com
作者简介:李志强(1981—), 男, 博士, 研究方向为分数阶偏微分方程的数值计算. E-mail: lizhiqiang0914@126.com
基金资助:
国家自然科学基金资助项目(11926319);山西省自然科学基金资助项目(201801D121010)

Asymptotics and large time behavior of solutions to a type of time-space fractional wave equation

LI Zhiqiang¹^,²()

1. College of Sciences, Shanghai University, Shanghai 200444, China
2. Department of Mathematics, Lüliang University, Lüliang 033001, Shanxi, China

Received:2020-05-08 Online:2021-12-31 Published:2020-09-23
Contact: LI Zhiqiang E-mail:lizhiqiang0914@126.com

摘要/Abstract

摘要：

研究带分数阶 Laplace 算子的时间-空间分数阶偏微分方程解的渐近性, 其中时间分数阶导数是在 Caputo 导数意义下, 其导数阶 $\alpha\in(1,2)$. 利用 Fox $H$-函数的性质和 Young 不等式给出了解的梯度估计, 并且研究了其长时间行为.

关键词: Caputo 导数, 分数阶 Laplace 算子, 渐近性, 长时间行为

Abstract:

This study investigates the asymptotic behaviors of a solution to time-space fractional partial differential equation with the fractional Laplacian, where the time fractional derivative is in the sense of Caputo, with the order $\alpha\in(1,2)$. By using the properties of the Fox $H$-function and Young's inequality, gradient estimates and large time behavior of the solution are obtained.

Key words: Caputo derivative, fractional Laplacian, asymptotic behavior, large time behavior

中图分类号:

O175.27

李志强. 时空分数阶波方程解的渐近性与长时间行为[J]. 上海大学学报(自然科学版), 2021, 27(6): 1149-1161.

LI Zhiqiang. Asymptotics and large time behavior of solutions to a type of time-space fractional wave equation[J]. Journal of Shanghai University（Natural Science Edition）, 2021, 27(6): 1149-1161.

参考文献 13

[1]	Podlubny I. Fractional differential equations [M]. New York: Academic Press, 1999: 1-198.
[2]	Kilbas A A, Srivastava H M, Trujillo J J. Theory and applications of fractional differential equations[M]. Amsterdam: Elsevier Science, 2006: 1-388.
[3]	Li C P, Zeng F H. Numerical methods for fractional calculus[M]. Boca Raton: Chapman and Hall/CRC, 2015: 1-27.
[4]	Li C P, Cai M. Theory and numerical approximations of fractional integrals and derivatives[M]. Philadelphia: SIAM, 2019: 1-71.
[5]	Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces[J]. Bull Sci Math, 2012, 136:521-573. doi: 10.1016/j.bulsci.2011.12.004
[6]	Ma Y T, Zhang F R, Li C P. The asymptotics of the solutions to the anomalous diffusion equations[J]. Comput Math Appl, 2013, 66(5): 682-692. doi: 10.1016/j.camwa.2013.01.032
[7]	Kemppainen J, Siljander J, Vergara V, et al. Decay estimates for time-fractional and other non-local in time subdiffusion equations in ${\mathbf R}^{d}$[J]. Math Ann, 2016, 366(3): 941-979. doi: 10.1007/s00208-015-1356-z
[8]	Kemppainen J, Siljander J, Zacher R. Representation of solutions and large-time behavior for fully nonlocal diffusion equations[J]. J Diff Equ, 2017, 263(1): 149-201. doi: 10.1016/j.jde.2017.02.030
[9]	Li C P, Li Z Q. Asymptotic behaviors of solution to Caputo-Hadamard fractional partial differential equation with fractional Laplacian[J]. Int J Comput Math, 2020, 98(2): 305-339. doi: 10.1080/00207160.2020.1744574
[10]	Djida J D, Fernandez A, Area I. Well-posedness results for fractional semi-linear wave equations[J]. Discrete Continuous Dynam. Systems-B, 2020, 25(2): 569-597.
[11]	Kilbas A A, Saigo M. $H$-transforms: theory and applications[M]. Boca Raton: CRC Press, 2004: 1-40.
[12]	Srivastava H M, Gupta K C, Goyal S P. The $H$-functions of one and two variables with applications[M]. New Delhi: South Asian Publishers, 1982: 1-30.
[13]	Grafakos L. Classical and modern Fourier analysis[M]. London: Pearson Education, 2004: 16-60.

时空分数阶波方程解的渐近性与长时间行为

Asymptotics and large time behavior of solutions to a type of time-space fractional wave equation

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献 13

相关文章 3

编辑推荐

Metrics

本文评价

[1]	周长亮,王远弟. 一类拟线性抛物型方程的非局部边值问题[J]. 上海大学学报(自然科学版), 2011, 17(5): 606-613.
[2]	马丹,何斌吾. lⁿ_p空间中单位球的距离及其应用[J]. 上海大学学报(自然科学版), 2009, 15(1): 14-16.
[3]	薛莲;陈巧云;何斌吾. B_pⁿ的迷向常数及其渐近性质[J]. 上海大学学报(自然科学版), 2008, 14(3): 260-264 .