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doi: 10.3934/ipi.2021040
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## Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates

 1 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan 2 Department of Liberal Arts and Sciences, Faculty of Engineering, Takushoku University, Tatemachi, Hachioji, Tokyo, 193-0985, Japan

* Corresponding author: Xinchi Huang

Received  October 2020 Revised  March 2021 Early access May 2021

Fund Project: The first author is supported by Japan Society for the Promotion of Science under the program of JSPS Postdoctoral Fellowships for Research in Japan

We consider a half-order time-fractional diffusion equation in arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some additional assumptions. We establish the stability estimate of Lipschitz type in the inverse problems and the proofs are based on the Bukhgeim-Klibanov method by using Carleman estimates.

Citation: Xinchi Huang, Atsushi Kawamoto. Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates. Inverse Problems & Imaging, doi: 10.3934/ipi.2021040
##### References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resources Research, 28 (1992), 3293-3307.  doi: 10.1029/92WR01757.  Google Scholar [2] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar [3] O. Y. Emanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900.  doi: 10.1070/sm1995v186n06abeh000047.  Google Scholar [4] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, Vol. 34, Seoul National University, Seoul, 1996.  Google Scholar [5] S. Guerrero and K. Kassab, Carleman estimate and null controllability of a fourth order parabolic equation in dimension $N\geq2$, J. Math. Pures Appl. (9), 121 (2019), 135-161.  doi: 10.1016/j.matpur.2018.04.004.  Google Scholar [6] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research, 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.  Google Scholar [7] X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Probl., 35 (2019), 045003. doi: 10.1088/1361-6420/ab0138.  Google Scholar [8] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Probl., 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar [9] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. doi: 10.1007/0-387-32183-7.  Google Scholar [10] A. Kawamoto, Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates, J. Inverse Ill-Posed Probl., 26 (2018), 647-672.  doi: 10.1515/jiip-2016-0029.  Google Scholar [11] A. Kawamoto and M. Machida, Lipschitz stability in inverse source and inverse coefficient problems for a first- and half-order time-fractional diffusion equation, SIAM J. Math. Anal., 52 (2020), 967-1005.  doi: 10.1137/18M1235776.  Google Scholar [12] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Probl., 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [13] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.  Google Scholar [14] M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, in Inverse and Ill-Posed Problems Series, Vol. 46, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.  Google Scholar [15] Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications. Vol. 2: Fractional Differential Equations, De Gruyter, Berlin, (2019), 411–430. doi: 10.1515/9783110571660-018.  Google Scholar [16] Z. Li and M. Yamamoto, Inverse problems of determining coefficients of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications. Vol.2: Fractional Differential Equations, De Gruyter, Berlin, (2019), 443–464. doi: 10.1515/9783110571660-020.  Google Scholar [17] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar [18] C. Ren and X. Xu, Local stability for an inverse coefficient problem of a fractional diffusion equation, Chin. Ann. Math., Ser. B, 35 (2014), 429-446.  doi: 10.1007/s11401-014-0833-0.  Google Scholar [19] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [20] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for fractional diffusion equation with half order and application, Appl. Anal., 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199.  Google Scholar [21] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar [22] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Probl., 28 (2012), 105010. doi: 10.1088/0266-5611/28/10/105010.  Google Scholar

show all references

##### References:
 [1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resources Research, 28 (1992), 3293-3307.  doi: 10.1029/92WR01757.  Google Scholar [2] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269-272.   Google Scholar [3] O. Y. Emanuvilov, Controllability of parabolic equations, Sbornik Math., 186 (1995), 879-900.  doi: 10.1070/sm1995v186n06abeh000047.  Google Scholar [4] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, Vol. 34, Seoul National University, Seoul, 1996.  Google Scholar [5] S. Guerrero and K. Kassab, Carleman estimate and null controllability of a fourth order parabolic equation in dimension $N\geq2$, J. Math. Pures Appl. (9), 121 (2019), 135-161.  doi: 10.1016/j.matpur.2018.04.004.  Google Scholar [6] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research, 34 (1998), 1027-1033.  doi: 10.1029/98WR00214.  Google Scholar [7] X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Probl., 35 (2019), 045003. doi: 10.1088/1361-6420/ab0138.  Google Scholar [8] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Probl., 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar [9] V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 2006. doi: 10.1007/0-387-32183-7.  Google Scholar [10] A. Kawamoto, Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates, J. Inverse Ill-Posed Probl., 26 (2018), 647-672.  doi: 10.1515/jiip-2016-0029.  Google Scholar [11] A. Kawamoto and M. Machida, Lipschitz stability in inverse source and inverse coefficient problems for a first- and half-order time-fractional diffusion equation, SIAM J. Math. Anal., 52 (2020), 967-1005.  doi: 10.1137/18M1235776.  Google Scholar [12] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Probl., 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.  Google Scholar [13] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), 477-560.  doi: 10.1515/jip-2012-0072.  Google Scholar [14] M. V. Klibanov and A. A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, in Inverse and Ill-Posed Problems Series, Vol. 46, VSP, Utrecht, 2004. doi: 10.1515/9783110915549.  Google Scholar [15] Y. Liu, Z. Li and M. Yamamoto, Inverse problems of determining sources of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications. Vol. 2: Fractional Differential Equations, De Gruyter, Berlin, (2019), 411–430. doi: 10.1515/9783110571660-018.  Google Scholar [16] Z. Li and M. Yamamoto, Inverse problems of determining coefficients of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications. Vol.2: Fractional Differential Equations, De Gruyter, Berlin, (2019), 443–464. doi: 10.1515/9783110571660-020.  Google Scholar [17] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar [18] C. Ren and X. Xu, Local stability for an inverse coefficient problem of a fractional diffusion equation, Chin. Ann. Math., Ser. B, 35 (2014), 429-446.  doi: 10.1007/s11401-014-0833-0.  Google Scholar [19] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382 (2011), 426-447.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [20] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for fractional diffusion equation with half order and application, Appl. Anal., 90 (2011), 1355-1371.  doi: 10.1080/00036811.2010.507199.  Google Scholar [21] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Probl., 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar [22] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Probl., 28 (2012), 105010. doi: 10.1088/0266-5611/28/10/105010.  Google Scholar
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