Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates

References

[1] E. E. Adams and L. W. Gelhar, Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis, Water Resources Res. 28 (1992), 3293–3307. 10.1029/92WR01757Suche in Google Scholar

[2] T. S. Aleroev, M. Kirane and S. A. Malik, Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition, Electron. J. Differential Equations 270 (2013), Article ID 270. Suche in Google Scholar

[3] L. Baudouin, E. Cerpa, E. Crépeau and A. Mercado, Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation, Appl. Anal. 92 (2013), no. 10, 2084–2102. 10.1080/00036811.2012.716589Suche in Google Scholar

[4] A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems (in Russian), Dokl. Akad. Nauk SSSR 260 (1981), no. 2, 269–272. Suche in Google Scholar

[5] N. Carreño and P. Guzmán, On the cost of null controllability of a fourth-order parabolic equation, J. Differential Equations 261 (2016), no. 11, 6485–6520. 10.1016/j.jde.2016.08.042Suche in Google Scholar

[6] E. Cerpa, P. Guzmán and A. Mercado, On the control of the linear Kuramoto–Sivashinsky equation, ESAIM Control Optim. Calc. Var. 23 (2017), no. 1, 165–194. 10.1051/cocv/2015044Suche in Google Scholar

[7] E. Cerpa and A. Mercado, Local exact controllability to the trajectories of the 1-D Kuramoto–Sivashinsky equation, J. Differential Equations 250 (2011), no. 4, 2024–2044. 10.1016/j.jde.2010.12.015Suche in Google Scholar

[8] J. Cheng, C.-L. Lin and G. Nakamura, Unique continuation property for the anomalous diffusion and its application, J. Differential Equations 254 (2013), no. 9, 3715–3728. 10.1016/j.jde.2013.01.039Suche in Google Scholar

[9] K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, Math. Control Relat. Fields 6 (2016), no. 2, 251–269. 10.3934/mcrf.2016003Suche in Google Scholar

[10] P. Gao, Insensitizing controls for the Cahn–Hilliard type equation, Electron. J. Qual. Theory Differ. Equ. 2014 (2014), Paper No. 35. 10.14232/ejqtde.2014.1.35Suche in Google Scholar

[11] P. Gao, A new global Carleman estimate for the one-dimensional Kuramoto–Sivashinsky equation and applications to exact controllability to the trajectories and an inverse problem, Nonlinear Anal. 117 (2015), 133–147. 10.1016/j.na.2015.01.015Suche in Google Scholar

[12] P. Gao, A new global Carleman estimate for Cahn–Hilliard type equation and its applications, J. Differential Equations 260 (2016), no. 1, 427–444. 10.1016/j.jde.2015.08.053Suche in Google Scholar

[13] P. Gao, Local exact controllability to the trajectories of the Swift–Hohenberg equation, Nonlinear Anal. 139 (2016), 169–195. 10.1016/j.na.2016.02.023Suche in Google Scholar

[14] P. Gao, M. Chen and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto–Sivashinsky equations, SIAM J. Control Optim. 53 (2015), no. 1, 475–500. 10.1137/130943820Suche in Google Scholar

[15] P. Guzmán Meléndez, Lipschitz stability in an inverse problem for the main coefficient of a Kuramoto–Sivashinsky type equation, J. Math. Anal. Appl. 408 (2013), no. 1, 275–290. 10.1016/j.jmaa.2013.05.050Suche in Google Scholar

[16] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Res. 34 (1998), 1027–1033. 10.1029/98WR00214Suche in Google Scholar

[17] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems 14 (1998), no. 5, 1229–1245. 10.1088/0266-5611/14/5/009Suche in Google Scholar

[18] J. Jia, J. Peng and J. Yang, Harnack’s inequality for a space-time fractional diffusion equation and applications to an inverse source problem, J. Differential Equations 262 (2017), no. 8, 4415–4450. 10.1016/j.jde.2017.01.002Suche in Google Scholar

[19] 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), no. 5, Article ID 055013. 10.1088/1361-6420/aa58d1Suche in Google Scholar

[20] B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28 (2012), no. 7, Article ID 075010. 10.1088/0266-5611/28/7/075010Suche in Google Scholar

[21] M. Kirane, S. A. Malik and M. A. Al-Gwaiz, An inverse source problem for a two-dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Methods Appl. Sci. 36 (2013), no. 9, 1056–1069. 10.1002/mma.2661Suche in Google Scholar

[22] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems 8 (1992), no. 4, 575–596. 10.1088/0266-5611/8/4/009Suche in Google Scholar

[23] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl. 21 (2013), no. 4, 477–560. 10.1515/jip-2012-0072Suche in Google Scholar

[24] M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2004. 10.1515/9783110915549Suche in Google Scholar

[25] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems 29 (2013), no. 6, Article ID 065014. 10.1088/0266-5611/29/6/065014Suche in Google Scholar

[26] C.-L. Lin and G. Nakamura, Unique continuation property for anomalous slow diffusion equation, Comm. Partial Differential Equations 41 (2016), no. 5, 749–758. 10.1080/03605302.2015.1135164Suche in Google Scholar

[27] Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem, Comput. Math. Appl. 73 (2017), no. 1, 96–108. 10.1016/j.camwa.2016.10.021Suche in Google Scholar

[28] 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), no. 4, 888–906. 10.1515/fca-2016-0048Suche in Google Scholar

[29] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1–77. 10.1016/S0370-1573(00)00070-3Suche in Google Scholar

[30] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems 29 (2013), no. 7, Article ID 075013. 10.1088/0266-5611/29/7/075013Suche in Google Scholar

[31] C. Ren and X. Xu, Local stability for an inverse coefficient problem of a fractional diffusion equation, Chin. Ann. Math. Ser. B 35 (2014), no. 3, 429–446. 10.1007/s11401-014-0833-0Suche in Google Scholar

[32] Z. Ruan and Z. Wang, Identification of a time-dependent source term for a time fractional diffusion problem, Appl. Anal. 96 (2017), no. 10, 1638–1655. 10.1080/00036811.2016.1232400Suche in Google Scholar

[33] 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), no. 1, 426–447. 10.1016/j.jmaa.2011.04.058Suche in Google Scholar

[34] K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields 1 (2011), no. 4, 509–518. 10.3934/mcrf.2011.1.509Suche in Google Scholar

[35] L. Sun and T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation, Appl. Numer. Math. 111 (2017), 160–180. 10.1016/j.apnum.2016.09.005Suche in Google Scholar

[36] S. Tatar and S. Ulusoy, An inverse source problem for a one-dimensional space-time fractional diffusion equation, Appl. Anal. 94 (2015), no. 11, 2233–2244. 10.1080/00036811.2014.979808Suche in Google Scholar

[37] V. K. Tuan, Inverse problem for fractional diffusion equation, Fract. Calc. Appl. Anal. 14 (2011), no. 1, 31–55. 10.2478/s13540-011-0004-xSuche in Google Scholar

[38] J.-G. Wang, Y.-B. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math. 68 (2013), 39–57. 10.1016/j.apnum.2013.01.001Suche in Google Scholar

[39] T. Wei, X. L. Li and Y. S. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Problems 32 (2016), no. 8, Article ID 085003. 10.1088/0266-5611/32/8/085003Suche in Google Scholar

[40] T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math. 78 (2014), 95–111. 10.1016/j.apnum.2013.12.002Suche in Google Scholar

[41] B. Wu and S. Wu, Existence and uniqueness of an inverse source problem for a fractional integrodifferential equation, Comput. Math. Appl. 68 (2014), no. 10, 1123–1136. 10.1016/j.camwa.2014.08.014Suche in Google Scholar

[42] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for a fractional diffusion equation with half order and application, Appl. Anal. 90 (2011), no. 9, 1355–1371. 10.1080/00036811.2010.507199Suche in Google Scholar

[43] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, Article ID 123013. 10.1088/0266-5611/25/12/123013Suche in Google Scholar

[44] 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 Problems 28 (2012), no. 10, Article ID 105010. 10.1088/0266-5611/28/10/105010Suche in Google Scholar

[45] F. Yang, C.-L. Fu and X.-X. Li, The inverse source problem for time-fractional diffusion equation: Stability analysis and regularization, Inverse Probl. Sci. Eng. 23 (2015), no. 6, 969–996. 10.1080/17415977.2014.968148Suche in Google Scholar

[46] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems 27 (2011), no. 3, Article ID 035010. 10.1088/0266-5611/27/3/035010Suche in Google Scholar

[47] Z. Zhou, Observability estimate and null controllability for one-dimensional fourth order parabolic equation, Taiwanese J. Math. 16 (2012), no. 6, 1991–2017. 10.11650/twjm/1500406835Suche in Google Scholar