Inverse problems for a generalized subdiffusion equation with final overdetermination. (English) Zbl 1472.35456
Summary: We consider two inverse problems for a generalized subdiffusion equation that use the final overdetermination condition. Firstly, we study a problem of reconstruction of a specific space-dependent component in a source term. We prove existence, uniqueness and stability of the solution to this problem. Based on these results, we consider an inverse problem of identification of a space-dependent coefficient of a linear reaction term. We prove the uniqueness and local existence and stability of the solution to this problem.
References:
| [1] | E. Bajlekova. Fractional evolution equations in Banach spaces. PhD thesis, Technische Universiteit Eindhoven, Eindhoven, 1 2001. · Zbl 0989.34002 |
| [2] | E. Beretta and C. Cavaterra. Identifying a space dependent coefficient in a reaction-diffusion equation. Inverse Problems and Imaging, 5(2):285-296, 2011. https://doi.org/10.3934/ipi.2011.5.285. · Zbl 1219.35351 · doi:10.3934/ipi.2011.5.285 |
| [3] | A.V. Chechkin, V.Yu. Gonchar, R. Gorenflo, N. Korabel and I.M. Sokolov. Generalized fractional diffusion equations for accelerating subd-iffusion and truncated Lévy flights. Physical Review E, 78:021111, 2008. https://doi.org/10.1103/PhysRevE.78.021111. · doi:10.1103/PhysRevE.78.021111 |
| [4] | A.V. Chechkin, R. Gorenflo and I.M. Sokolov. Fractional diffusion in inhomogeneous media. J. Phys. A: Math. Gen., 38(42):679-684, 2005. https://doi.org/10.1088/0305-4470/38/42/L03. · Zbl 1082.76097 · doi:10.1088/0305-4470/38/42/L03 |
| [5] | P. Clement and J. Prüss. Completely positive measures and Feller semigroups. Mathematische Annalen, 287(1):73-105, 1990. https://doi.org/10.1007/BF01446879. · Zbl 0717.47013 · doi:10.1007/BF01446879 |
| [6] | D. Frömberg. Reaction Kinetics under Anomalous Diffusion. PhD thesis, Humboldt-Universität zu Berlin, Berlin, 8 1981. |
| [7] | K.M. Furati, O.S. Iyiola and M. Kirane. An inverse problem for a generalized fractional diffusion. Applied Mathematics and Computation, 249:24-31, 2014. https://doi.org/10.1016/j.amc.2014.10.046. · Zbl 1338.35493 · doi:10.1016/j.amc.2014.10.046 |
| [8] | D. Gilbarg and N. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1977. https://doi.org/10.1007/978-3-642-96379-7. · Zbl 0361.35003 · doi:10.1007/978-3-642-96379-7 |
| [9] | G. Gripenberg. On Volterra equations of the first kind. Integral Equations and Operator Theory, 3(4):473-488, 1980. https://doi.org/10.1007/BF01702311. · Zbl 0445.45001 · doi:10.1007/BF01702311 |
| [10] | G. Gripenberg, S.-O. Londen and O. Staffans. Volterra Integral and Functional Equations. Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511662805. · Zbl 0695.45002 · doi:10.1017/CBO9780511662805 |
| [11] | G.H. Hardy and J.E. Littlewood. Some properties of fractional integrals I. Math-ematische Zeitschrift, 27:565-606, 1928. https://doi.org/10.1007/BF01171116. · JFM 54.0275.05 · doi:10.1007/BF01171116 |
| [12] | V. Isakov. Inverse Problems for Partial Differential Equations. Springer, New York, 2006. · Zbl 1092.35001 |
| [13] | J. Janno and K. Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems and Imaging, 3(1):17-41, 2009. https://doi.org/10.3934/ipi.2009.3.17. · Zbl 1187.35267 · doi:10.3934/ipi.2009.3.17 |
| [14] | J. Janno and K. Kasemets. Identification of a kernel in an evolutionary integral equation occurring in subdiffusion. Journal of Inverse and Ill-posed Problems, 25(6):777-798, 2017. https://doi.org/10.1515/jiip-2016-0082. · Zbl 1386.35476 · doi:10.1515/jiip-2016-0082 |
| [15] | J. Janno and K. Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems and Imaging, 11(1):125-149, 2017. https://doi.org/10.3934/ipi.2017007. · Zbl 1357.35296 · doi:10.3934/ipi.2017007 |
| [16] | K. Kasemets and J. Janno. Reconstruction of a source term in a parabolic integro-differential equation from final data. Mathematical Modelling and Anal-ysis, 16(2):199-219, 2011. https://doi.org/10.3846/13926292.2011.578282. · Zbl 1219.35362 · doi:10.3846/13926292.2011.578282 |
| [17] | N. Kinash and J. Janno. Inverse problems for a perturbed time fractional diffu-sion equation with final overdetermination. Math. Meth. Appl. Sci., 41(5):1925-1943, 2018. https://doi.org/10.1002/mma.4719. · Zbl 1391.35420 · doi:10.1002/mma.4719 |
| [18] | M. Kirane, S.A. Malik and M.A. Al-Gwaiz. An inverse source prob-lem for a two dimensional time fractional diffusion equation with nonlo-cal boundary conditions. Math. Meth. Appl. Sci., 36(9):1056-1069, 2013. https://doi.org/10.1002/mma.2661. · Zbl 1267.80013 · doi:10.1002/mma.2661 |
| [19] | A.N. Kochubei. General fractional calculus, evolution equations, and re-newal processes. Integral Equations and Operator Theory, 71(4):583-600, 2011. https://doi.org/10.1007/s00020-011-1918-8. · Zbl 1250.26006 · doi:10.1007/s00020-011-1918-8 |
| [20] | Y. Liu, M. Yamamoto and W. Rundell. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fractional Calculus and Applied Analysis, 19(4), 2016. https://doi.org/10.1515/fca-2016-0048. · Zbl 1346.35216 · doi:10.1515/fca-2016-0048 |
| [21] | A. Lopushansy and H. Lopushanska. Inverse source Cauchy problem for a time fractional diffusion-wave equation with distributions. Electronic Journal of Dif-ferential Equations, 2017(182):1-14, 2017. · Zbl 1370.35282 |
| [22] | Y. Luchko. Maximum principle and its application for the time-fractional diffu-sion equations. Fractional Calculus and Applied Analysis, 14(1):110-124, 2011. https://doi.org/10.2478/s13540-011-0008-6. · Zbl 1273.35297 · doi:10.2478/s13540-011-0008-6 |
| [23] | Y. Luchko and M. Yamamoto. General time-fractional diffusion equa-tion: Some uniqueness and existence results for the initial-boundary-value problems. Fractional Calculus and Applied Analysis, 19(3):676-695, 2016. https://doi.org/10.1515/fca-2016-0036. · Zbl 1499.35667 · doi:10.1515/fca-2016-0036 |
| [24] | A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995. · Zbl 0816.35001 |
| [25] | F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo. Time-fractional diffusion of distributed order. Journal of Vibration and Control, 14(9-10):1267-1290, 2008. https://doi.org/10.1177/1077546307087452. · Zbl 1229.35118 · doi:10.1177/1077546307087452 |
| [26] | D.G. Orlovsky. Parameter determination in a differential equation of fractional order with Riemann-Liouville fractional derivative in a Hilbert space. Journal of Siberian Federal University, 8(1):55-63, 2015. https://doi.org/10.17516/1997-1397-2015-8-1-55-63. · Zbl 1525.34028 · doi:10.17516/1997-1397-2015-8-1-55-63 |
| [27] | Y. Povstenko. Fractional heat conduction and associated ther-mal stress. Journal of Thermal Stresses, 28(1):83-102, 2004. https://doi.org/10.1080/014957390523741. · doi:10.1080/014957390523741 |
| [28] | J. Prüss. Evolutionary Integral Equations and Applications. Birkhäuser Verlag, Basel, 1993. https://doi.org/10.1007/978-3-0348-8570-6. · Zbl 0793.45014 · doi:10.1007/978-3-0348-8570-6 |
| [29] | F. Sabzikar, M.M. Meerschaert and J. Chen. Tempered frac-tional calculus. Journal of Computational Physics, 293:14-28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024. · Zbl 1349.26017 · doi:10.1016/j.jcp.2014.04.024 |
| [30] | K. Sakamoto and M. Yamamoto. Inverse source problem with a final overde-termination for a fractional diffusion equation. Mathematical Control & Related Fields, 1(4):509-518, 2011. https://doi.org/10.3934/mcrf.2011.1.509. · Zbl 1241.35220 · doi:10.3934/mcrf.2011.1.509 |
| [31] | I.M. Sokolov, A.V. Chechkin and J. Klafter. Distributed-order fractional kinetics. Acta Physica Polonica B, 35(47):1323-1341, 2004. |
| [32] | S. Tatar and S. Ulusoy. An inverse source problem for a one-dimensional space-time fractional diffusion equation. Applicable Analysis, 94(11):2233-2244, 2015. https://doi.org/10.1080/00036811.2014.979808. · Zbl 1327.35416 · doi:10.1080/00036811.2014.979808 |
| [33] | V. Vergara and R. Zacher. Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations. Journal of Evolu-tion Equations, 17(1):599-626, 2017. https://doi.org/10.1007/s00028-016-0370-2. · Zbl 1365.35218 · doi:10.1007/s00028-016-0370-2 |
| [34] | R. Zacher. Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj, 52(1):1-18, 2009. https://doi.org/10.1619/fesi.52.1. · Zbl 1171.45003 · doi:10.1619/fesi.52.1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
