# American Institute of Mathematical Sciences

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. Cheng, C.-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. Gorenflo, Y. 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. Kubica, K. 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. Liu, W. 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. Xu, J. 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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##### 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. Cheng, C.-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. Gorenflo, Y. 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. Kubica, K. 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. Liu, W. 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. Xu, J. 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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