October  2021, 14(10): 3659-3683. doi: 10.3934/dcdss.2021023

Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case

Department of Mathematics, Shanghai University, 200444, Shanghai, China

* Corresponding author: Changpin Li

Received  April 2020 Revised  August 2020 Published  October 2021 Early access  March 2021

Fund Project: The first author is supported by NSFC grant 11872234 and 11926319

This paper is concerned with the asymptotic behaviors of solution to time–space fractional partial differential equation with Caputo–Hadamard derivative (in time) and fractional Laplacian (in space) in the hyperbolic case, that is, the Caputo–Hadamard derivative order $ \alpha $ lies in $ 1<\alpha<2 $. In view of the technique of integral transforms, the fundamental solutions and the exact solution of the considered equation are derived. Furthermore, the fundamental solutions are estimated and asymptotic behaviors of its analytical solution is established in $ L^{p}(\mathbb{R}^{d}) $ and $ L^{p,\infty} (\mathbb{R}^{d}) $. We finally investigate gradient estimates and large time behavior for the solution.

Citation: Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3659-3683. doi: 10.3934/dcdss.2021023
References:
[1]

B. Ahmad, A. Alsaedi, S. K. Ntouyas and J. Tariboon, Hadamard–Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Switzerland, 2017. doi: 10.1007/978-3-319-52141-1.  Google Scholar

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D. Baleanu and B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos, Solitons & Fractals, 116 (2018), 136-145.  doi: 10.1016/j.chaos.2018.09.020.  Google Scholar

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D. BaleanuB. ShiriH. M. Srivastava and M. AI Qurashi, A Chebyshev spectral method based on the operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Adv. Differ. Equ., 2018 (2018), 353-376.  doi: 10.1186/s13662-018-1822-5.  Google Scholar

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J.-D. DjidaA. Fernandez and I. Area, Well–posedness results for fractional semi-linear wave equations, Discrete Cont. Dyn.–B, 25 (2020), 569-597.  doi: 10.3934/dcdsb.2019255.  Google Scholar

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M. GoharC. P. Li and C. T. Yin, On Caputo-Hadamard fractional differential equations, Int. J. Comput. Math., 97 (2020), 1459-1483.  doi: 10.1080/00207160.2019.1626012.  Google Scholar

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M. GoharC. P. Li and Z. Q. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194-220.  doi: 10.1007/s00009-020-01605-4.  Google Scholar

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J. Hadamard, Essai sur létude des fonctions données par leur développement de Taylor, J. Math. Pures Appl., 8 (1892), 101-186.   Google Scholar

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Y. HuC. P. Li and H. F. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case, Chaos, Solitons & Fractals, 102 (2017), 319-326.  doi: 10.1016/j.chaos.2017.03.038.  Google Scholar

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Y. HuC. P. Li and H. F. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: More than one space dimension, Int. J. Comput. Math., 95 (2018), 1114-1130.  doi: 10.1080/00207160.2017.1378810.  Google Scholar

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F. JaradT. Abdeljawad and D. Baleanu, Caputo–type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142-150.  doi: 10.1186/1687-1847-2012-142.  Google Scholar

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A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.  Google Scholar

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K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion–wave equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

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J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time–fractional and other non-local in time subdiffusion equations in $\mathbb{R}^{d}$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar

[22]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large–time behavior for fully nonlocal diffusion equations, J. Diff. Equ., 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[23]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[24]

E. D. KhiabaniH. GhaffarzadehB. Shiri and J. Katebi, Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models, J. Vib. Control, 26 (2020), 1445-1462.  doi: 10.1177/1077546319898570.  Google Scholar

[25]

C. P. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, SIAM, Philadelphia, 2020.  Google Scholar

[26]

C. P. Li and Z. Q. Li, Asymptotic behaviors of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian, Int. J. Comput. Math., 98 (2021), 305-339. Google Scholar

[27]

C. P. Li, Z. Q. Li and Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation, J. Sci. Comput., 85 (2020), article 41. doi: 10.1007/s10915-020-01353-3.  Google Scholar

[28]

Y. T. MaF. R. Zhang and C. P. Li, The asymptotics of the solutions to the anomalous diffusion equations, Comput. Math. Appl., 66 (2013), 682-692.  doi: 10.1016/j.camwa.2013.01.032.  Google Scholar

[29]

C. Mou and Y. Yi, Interior regularity for regional fractional Laplacian, Comm. Math. Phys., 340 (2015), 233-251.  doi: 10.1007/s00220-015-2445-2.  Google Scholar

[30] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.   Google Scholar
[31]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures. Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[32]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar

[33]

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The $H$-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982.  Google Scholar

show all references

References:
[1]

B. Ahmad, A. Alsaedi, S. K. Ntouyas and J. Tariboon, Hadamard–Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Switzerland, 2017. doi: 10.1007/978-3-319-52141-1.  Google Scholar

[2]

N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann. Inst. H. Poincare Anal. Non Lineaire, 34 (2017), 439-467.  doi: 10.1016/j.anihpc.2016.02.001.  Google Scholar

[3]

B. L. J. Braaksma, Asymptotic expansions and analytical continuations for a class of Barnes–integrals, Compos. Math., 15 (1964), 239-341.   Google Scholar

[4]

D. Baleanu and B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos, Solitons & Fractals, 116 (2018), 136-145.  doi: 10.1016/j.chaos.2018.09.020.  Google Scholar

[5]

D. BaleanuB. ShiriH. M. Srivastava and M. AI Qurashi, A Chebyshev spectral method based on the operational matrix for fractional differential equations involving non-singular Mittag-Leffler kernel, Adv. Differ. Equ., 2018 (2018), 353-376.  doi: 10.1186/s13662-018-1822-5.  Google Scholar

[6]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[7]

J.-D. DjidaA. Fernandez and I. Area, Well–posedness results for fractional semi-linear wave equations, Discrete Cont. Dyn.–B, 25 (2020), 569-597.  doi: 10.3934/dcdsb.2019255.  Google Scholar

[8]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[9]

S. W. DuoH. Wang and Y. Z. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Cont. Dyn.–B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110.  Google Scholar

[10]

M. GoharC. P. Li and C. T. Yin, On Caputo-Hadamard fractional differential equations, Int. J. Comput. Math., 97 (2020), 1459-1483.  doi: 10.1080/00207160.2019.1626012.  Google Scholar

[11]

M. GoharC. P. Li and Z. Q. Li, Finite difference methods for Caputo-Hadamard fractional differential equations, Mediterr. J. Math., 17 (2020), 194-220.  doi: 10.1007/s00009-020-01605-4.  Google Scholar

[12]

L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, London, 2004.  Google Scholar

[13]

J. Hadamard, Essai sur létude des fonctions données par leur développement de Taylor, J. Math. Pures Appl., 8 (1892), 101-186.   Google Scholar

[14]

Y. HuC. P. Li and H. F. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: One-dimension case, Chaos, Solitons & Fractals, 102 (2017), 319-326.  doi: 10.1016/j.chaos.2017.03.038.  Google Scholar

[15]

Y. HuC. P. Li and H. F. Li, The finite difference method for Caputo-type parabolic equation with fractional Laplacian: More than one space dimension, Int. J. Comput. Math., 95 (2018), 1114-1130.  doi: 10.1080/00207160.2017.1378810.  Google Scholar

[16]

F. JaradT. Abdeljawad and D. Baleanu, Caputo–type modification of the Hadamard fractional derivatives, Adv. Differ. Equ., 2012 (2012), 142-150.  doi: 10.1186/1687-1847-2012-142.  Google Scholar

[17]

A. A. Kilbas, Hadamard–type fractional calculus, J. Korean Math. Soc., 38 (2001), 1191-1204.   Google Scholar

[18]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.  Google Scholar

[19] A. A. Kilbas and M. Saigo, $H$-Transforms: Theory and Applications, CRC Press, Boca Raton, 2004.  doi: 10.1201/9780203487372.  Google Scholar
[20]

K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion–wave equations, J. Korean Math. Soc., 53 (2016), 929-967.  doi: 10.4134/JKMS.j150343.  Google Scholar

[21]

J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time–fractional and other non-local in time subdiffusion equations in $\mathbb{R}^{d}$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.  Google Scholar

[22]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large–time behavior for fully nonlocal diffusion equations, J. Diff. Equ., 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.  Google Scholar

[23]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[24]

E. D. KhiabaniH. GhaffarzadehB. Shiri and J. Katebi, Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models, J. Vib. Control, 26 (2020), 1445-1462.  doi: 10.1177/1077546319898570.  Google Scholar

[25]

C. P. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, SIAM, Philadelphia, 2020.  Google Scholar

[26]

C. P. Li and Z. Q. Li, Asymptotic behaviors of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian, Int. J. Comput. Math., 98 (2021), 305-339. Google Scholar

[27]

C. P. Li, Z. Q. Li and Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation, J. Sci. Comput., 85 (2020), article 41. doi: 10.1007/s10915-020-01353-3.  Google Scholar

[28]

Y. T. MaF. R. Zhang and C. P. Li, The asymptotics of the solutions to the anomalous diffusion equations, Comput. Math. Appl., 66 (2013), 682-692.  doi: 10.1016/j.camwa.2013.01.032.  Google Scholar

[29]

C. Mou and Y. Yi, Interior regularity for regional fractional Laplacian, Comm. Math. Phys., 340 (2015), 233-251.  doi: 10.1007/s00220-015-2445-2.  Google Scholar

[30] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.   Google Scholar
[31]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures. Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[32]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.  Google Scholar

[33]

H. M. Srivastava, K. C. Gupta and S. P. Goyal, The $H$-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, 1982.  Google Scholar

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