Advanced Search
Article Contents
Article Contents

Inverse source problem for a one-dimensional time-fractional diffusion equation and unique continuation for weak solutions

  • *Corresponding author: Yikan Liu

    *Corresponding author: Yikan Liu 
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35R11, 35R30; Secondary: 35B53.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] R. A. AdamsSobolev 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. TrudingerElliptic 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. IsakovInverse 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. YamamotoTime-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. PodlubnyFractional Differential Equations, Academic Press, an Diego, 1999. 
    [21] W. RudinReal 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. ShakarchiComplex 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. YôsidaFunctional Analysis, $6^{th}$ edition, Springer, Berlin, 1980. 
  • 加载中

Article Metrics

HTML views(1257) PDF downloads(378) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint