doi: 10.3934/ipi.2022027
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Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions

1. 

School of Mathematics and Statistics, Ningbo University, 818 Fenghua Road, Ningbo, Zhejiang 315211, China

2. 

Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo 060-0812, Japan

3. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan

4. 

Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania

5. 

Correspondence member of Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1 98122 Messina, Italy

*Corresponding author: Yikan Liu

Received  December 2021 Revised  March 2022 Early access May 2022

In this paper, we obtain the sharp uniqueness for an inverse $ x $-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique continuation which holds for weak solutions.

Citation: Zhiyuan Li, Yikan Liu, Masahiro Yamamoto. Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions. Inverse Problems and Imaging, doi: 10.3934/ipi.2022027
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

E. E. Adams and L. W. Gelhar, Field study of dispersion in an heterogeneous aquifer 2. Spatial moments analysis, Water Resour. Res., 28 (1992), 3293-307.  doi: 10.1029/92WR01757.

[3]

J. ChengC.-L. Lin and G. Nakamura, Unique continuation property for the anomalous diffusion and its application, J. Differential Equations, 254 (2013), 3715-3728.  doi: 10.1016/j.jde.2013.01.039.

[4]

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

[5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. 
[6]

R. GorenfloY. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.

[7]

Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resour. Res., 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.

[8]

X, Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003, 36 pp. doi: 10.1088/1361-6420/ab0138.

[9] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2006. 
[10]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013, 22 pp. doi: 10.1088/1361-6420/aa58d1.

[11] A. KubicaK. Ryszewska and M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, Springer, Singapore, 2020. 
[12]

B. M. Levitan and I. S. Sargsian, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, AMS, Providence, 1975.

[13]

Z. Li and M. Yamamoto, Unique continuation principle for the one-dimensional time fractional diffusion equation, Fract. Calc. Appl. Anal., 22 (2019), 664-657.  doi: 10.1515/fca-2019-0036.

[14]

C.-L. Lin and G. Nakamura, Unique continuation property for anomalous slow diffusion equation, Comm. Partial Differential Equations, 41 (2016), 749-758.  doi: 10.1080/03605302.2015.1135164.

[15]

C.-L. Lin and G. Nakamura, Unique continuation property for multi-terms time fractional diffusion equations, Math. Ann., 373 (2019), 929-952.  doi: 10.1007/s00208-018-1710-z.

[16]

C.-L. Lin and G. Nakamura, Classical unique continuation property for multi-term time-fractional evolution equations, to appear, Math. Ann..

[17]

Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial differential equations, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,431–442.

[18]

Y. LiuW. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fract. Calc. Appl. Anal., 19 (2016), 888-906.  doi: 10.1515/fca-2016-0048.

[19]

Y. Luchko and M. Yamamoto, Maximum principle for the time-fractional PDEs, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,299–326.

[20] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. 
[21] W. Rudin, Real and Complex Analysis, McGraw-Hill, Osborne, 1974. 
[22]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[23]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.

[24] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, 2003. 
[25]

S. Umarov, Fractional Duhamel principle, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,383–410.,

[26]

X. XuJ. Cheng and M. Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application, Appl. Anal., 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199.

[27]

M. Yamamoto, Fractional calculus and time-fractional differential equations: revisit and construction of a theory, Math., 10 (2022), 698, 55 pp. doi: 10.3390/math10050698.

[28] K. Yôsida, Functional Analysis, $6^{th}$ edition, Springer, Berlin, 1980. 

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. 
[2]

E. E. Adams and L. W. Gelhar, Field study of dispersion in an heterogeneous aquifer 2. Spatial moments analysis, Water Resour. Res., 28 (1992), 3293-307.  doi: 10.1029/92WR01757.

[3]

J. ChengC.-L. Lin and G. Nakamura, Unique continuation property for the anomalous diffusion and its application, J. Differential Equations, 254 (2013), 3715-3728.  doi: 10.1016/j.jde.2013.01.039.

[4]

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

[5] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. 
[6]

R. GorenfloY. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048.

[7]

Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: an explanation of long-tailed profiles, Water Resour. Res., 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.

[8]

X, Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003, 36 pp. doi: 10.1088/1361-6420/ab0138.

[9] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer, Berlin, 2006. 
[10]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013, 22 pp. doi: 10.1088/1361-6420/aa58d1.

[11] A. KubicaK. Ryszewska and M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, Springer, Singapore, 2020. 
[12]

B. M. Levitan and I. S. Sargsian, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, AMS, Providence, 1975.

[13]

Z. Li and M. Yamamoto, Unique continuation principle for the one-dimensional time fractional diffusion equation, Fract. Calc. Appl. Anal., 22 (2019), 664-657.  doi: 10.1515/fca-2019-0036.

[14]

C.-L. Lin and G. Nakamura, Unique continuation property for anomalous slow diffusion equation, Comm. Partial Differential Equations, 41 (2016), 749-758.  doi: 10.1080/03605302.2015.1135164.

[15]

C.-L. Lin and G. Nakamura, Unique continuation property for multi-terms time fractional diffusion equations, Math. Ann., 373 (2019), 929-952.  doi: 10.1007/s00208-018-1710-z.

[16]

C.-L. Lin and G. Nakamura, Classical unique continuation property for multi-term time-fractional evolution equations, to appear, Math. Ann..

[17]

Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial differential equations, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,431–442.

[18]

Y. LiuW. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem, Fract. Calc. Appl. Anal., 19 (2016), 888-906.  doi: 10.1515/fca-2016-0048.

[19]

Y. Luchko and M. Yamamoto, Maximum principle for the time-fractional PDEs, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,299–326.

[20] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. 
[21] W. Rudin, Real and Complex Analysis, McGraw-Hill, Osborne, 1974. 
[22]

K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.

[23]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139.  doi: 10.1016/0022-0396(87)90043-X.

[24] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, Princeton, 2003. 
[25]

S. Umarov, Fractional Duhamel principle, Fractional Differential Equations, Handbook of Fractional Calculus with Applications, 2, De Gruyter, Berlin, 2019,383–410.,

[26]

X. XuJ. Cheng and M. Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application, Appl. Anal., 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199.

[27]

M. Yamamoto, Fractional calculus and time-fractional differential equations: revisit and construction of a theory, Math., 10 (2022), 698, 55 pp. doi: 10.3390/math10050698.

[28] K. Yôsida, Functional Analysis, $6^{th}$ edition, Springer, Berlin, 1980. 
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