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 and 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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

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

[32]

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

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[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.

[17]

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

[18]

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

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

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

[32]

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

[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.

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