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Journal of Shanghai University >

2021 , Vol. 27 >Issue 6: 1149 - 1161

DOI: https://doi.org/10.12066/j.issn.1007-2861.2257

Research Articles

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

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

Received date: 2020-05-08

  Online published: 2020-09-14

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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

Cite this article

LI Zhiqiang . Asymptotics and large time behavior of solutions to a type of time-space fractional wave equation[J]. Journal of Shanghai University, 2021 , 27(6) : 1149 -1161 . DOI: 10.12066/j.issn.1007-2861.2257

References

[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.
[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.
[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.
[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.
[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.
[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.
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