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  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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

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

[19]

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

[20]

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

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

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