July  2019, 24(7): 3195-3210. doi: 10.3934/dcdsb.2018340

Regularity of solutions to time fractional diffusion equations

School of Mathematics, Sichuan University, Chengdu 610064, China

* Corresponding author: Xiaoping Xie, xpxie@scu.edu.cn

Received  June 2017 Revised  December 2017 Published  July 2019 Early access  January 2019

Fund Project: This work was supported by National Natural Science Foundation of China (11771312) and Major Research Plan of National Natural Science Foundation of China (91430105).

We derive some regularity estimates of the solution to a time fractional diffusion equation by using the Galerkin method. The regularity estimates partially unravel the singularity structure of the solution with respect to the time variable. We show that the regularity of the weak solution can be improved by subtracting some particular forms of singular functions.

Citation: Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3195-3210. doi: 10.3934/dcdsb.2018340
References:
[1]

O. P. Agarwal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.  doi: 10.1023/A:1016539022492.

[2]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[3]

A. M. A. El-Sayed, Frartiorial order tliffusion-wave equation, International Journal of Theoretical Physics, 35 (1996), 311-322.  doi: 10.1007/BF02083817.

[4]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numerical Methods for Partial Differential Equations, 22 (2006), 558-576.  doi: 10.1002/num.20112.

[5]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 1998.

[6]

Z. Fan, Existence and regularity of solutions for evolution equations with Riemann-Liouville fractional derivatives, Indagationes Mathematicae, 25 (2014), 516-524.  doi: 10.1016/j.indag.2014.01.002.

[7]

V. D. Gejji and H. Jafari, Boundary value problems for fractional diffusion-wave equation, The Australian Journal of Mathematical Analysis and Applications, 3 (2006), Art. 16, 8 pp.

[8]

X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[9]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, Berlin Heidelberg, 1972.

[10]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223.  doi: 10.1016/j.jmaa.2008.10.018.

[11]

Y. Luchko, Some uniqueness and existence results for the initial-boundary value problems for the generalized time-fractional diffusion equation, Mathematics with Applications, 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015.

[12]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fractional Calculus and Applied Analysis, 15 (2012), 141-160.  doi: 10.2478/s13540-012-0010-7.

[13]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in: Waves and Stability in Continuous Media, World Scientific, Singapore, 23 (1994), 246-251.

[14]

F. Mainardi, The time fractional diffusion-wave equation, Radiophysics and Quantum Electronics, 38 (1995), 13-24.  doi: 10.1007/BF01051854.

[15]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics Letters, 9 (1996), 23-28.  doi: 10.1016/0893-9659(96)00089-4.

[16]

J. MuB. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.

[17] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[18]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equation and application to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[19]

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

[20]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer-Verlag Berlin Heidelberg, 2007.

[21]

R. WangD. Chen and T. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[22]

K. Yosida, Functional Analysis, sixth edition, Springer-Verlag, Berlin Heidelberg, 1980.

[23]

R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Mathematische Annalen, 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.

[24]

Z. Zhang and B. Liu, Existence of mild solutions for fractional evolution equations, Journal of Fractional Calculus and Applications, 2 (2012), 1-10. 

[25]

M. ZhengF. LiuI. Turner and V. Anh, A novel high order space-time spectral method for the time fractional Fokker-Planck equation, SIAM Journal on Scientific Computing, 37 (2015), 701-724.  doi: 10.1137/140980545.

show all references

References:
[1]

O. P. Agarwal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynamics, 29 (2002), 145-155.  doi: 10.1023/A:1016539022492.

[2]

S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, Journal of Differential Equations, 199 (2004), 211-255.  doi: 10.1016/j.jde.2003.12.002.

[3]

A. M. A. El-Sayed, Frartiorial order tliffusion-wave equation, International Journal of Theoretical Physics, 35 (1996), 311-322.  doi: 10.1007/BF02083817.

[4]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numerical Methods for Partial Differential Equations, 22 (2006), 558-576.  doi: 10.1002/num.20112.

[5]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 1998.

[6]

Z. Fan, Existence and regularity of solutions for evolution equations with Riemann-Liouville fractional derivatives, Indagationes Mathematicae, 25 (2014), 516-524.  doi: 10.1016/j.indag.2014.01.002.

[7]

V. D. Gejji and H. Jafari, Boundary value problems for fractional diffusion-wave equation, The Australian Journal of Mathematical Analysis and Applications, 3 (2006), Art. 16, 8 pp.

[8]

X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM Journal on Numerical Analysis, 47 (2009), 2108-2131.  doi: 10.1137/080718942.

[9]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, Berlin Heidelberg, 1972.

[10]

Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 351 (2009), 218-223.  doi: 10.1016/j.jmaa.2008.10.018.

[11]

Y. Luchko, Some uniqueness and existence results for the initial-boundary value problems for the generalized time-fractional diffusion equation, Mathematics with Applications, 59 (2010), 1766-1772.  doi: 10.1016/j.camwa.2009.08.015.

[12]

Y. Luchko, Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation, Fractional Calculus and Applied Analysis, 15 (2012), 141-160.  doi: 10.2478/s13540-012-0010-7.

[13]

F. Mainardi, On the initial value problem for the fractional diffusion-wave equation, in: Waves and Stability in Continuous Media, World Scientific, Singapore, 23 (1994), 246-251.

[14]

F. Mainardi, The time fractional diffusion-wave equation, Radiophysics and Quantum Electronics, 38 (1995), 13-24.  doi: 10.1007/BF01051854.

[15]

F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation, Applied Mathematics Letters, 9 (1996), 23-28.  doi: 10.1016/0893-9659(96)00089-4.

[16]

J. MuB. Ahmad and S. Huang, Existence and regularity of solutions to time-fractional diffusion equations, Computers & Mathematics with Applications, 73 (2017), 985-996.  doi: 10.1016/j.camwa.2016.04.039.

[17] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. 
[18]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equation and application to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[19]

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

[20]

L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Springer-Verlag Berlin Heidelberg, 2007.

[21]

R. WangD. Chen and T. Xiao, Abstract fractional Cauchy problems with almost sectorial operators, Journal of Differential Equations, 252 (2012), 202-235.  doi: 10.1016/j.jde.2011.08.048.

[22]

K. Yosida, Functional Analysis, sixth edition, Springer-Verlag, Berlin Heidelberg, 1980.

[23]

R. Zacher, A De Giorgi-Nash type theorem for time fractional diffusion equations, Mathematische Annalen, 356 (2013), 99-146.  doi: 10.1007/s00208-012-0834-9.

[24]

Z. Zhang and B. Liu, Existence of mild solutions for fractional evolution equations, Journal of Fractional Calculus and Applications, 2 (2012), 1-10. 

[25]

M. ZhengF. LiuI. Turner and V. Anh, A novel high order space-time spectral method for the time fractional Fokker-Planck equation, SIAM Journal on Scientific Computing, 37 (2015), 701-724.  doi: 10.1137/140980545.

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