# American Institute of Mathematical Sciences

2011, 1(4): 509-518. doi: 10.3934/mcrf.2011.1.509

## Inverse source problem with a final overdetermination for a fractional diffusion equation

 1 Mathematical Science & Technology Research Lab, Advanced Technology Research Laboratories, Technical Development Bureau, Nippon Steel Corporation, 20-1 Shintomi, Futtsu, Chiba 293-8511, Japan 2 Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914

Received  December 2010 Revised  May 2011 Published  November 2011

For a time fractional diffusion equation with source term, we discuss an inverse problem of determining a spatially varying function of the source by final overdetermining data. We prove that this inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants.
Citation: Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509
##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). [2] M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition,, Inverse Problems, 13 (1997), 19. doi: 10.1088/0266-5611/13/1/003. [3] M. Choulli and M. Yamamoto, Generic well-posedness of an inverse parabolic problem--the Hölder-space approach,, Inverse Problems, 12 (1996), 195. doi: 10.1088/0266-5611/12/3/002. [4] M. Choulli and M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameters,, J. Inverse Ill-Posed Problems, 7 (1999), 241. doi: 10.1515/jiip.1999.7.3.241. [5] M. Ginoa, S. Gerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials,, Physica A, 191 (1992), 449. doi: 10.1016/0378-4371(92)90566-9. [6] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, (1997), 223. [7] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lec. Notes in Math. 840, (1981). [8] K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem,, in, 422 (1993), 49. [9] V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185. doi: 10.1002/cpa.3160440203. [10] F. John, "Partial Differential Equations,", Springer-Verlag, (1982). [11] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976). [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations,", Elsevier, (2006). [13] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation,, Waves and Stability in Continuous Media, (1994), 246. [14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Appl. Math. Lett., 9 (1996), 23. doi: 10.1016/0893-9659(96)00089-4. [15] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics,, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, (1997), 291. [16] R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry,, Phys. Stat. Sol. B, 133 (1986), 425. doi: 10.1002/pssb.2221330150. [17] K. B. Oldham and J. Spanier, "The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order,", Academic Press, (1974). [18] I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999). [19] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000). [20] J. Prüss, "Evolutionary Integral Equations and Applications,", Birkhäuser, (1993). [21] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation,, J. Phys. A, 27 (1994), 3407. doi: 10.1088/0305-4470/27/10/017. [22] K. Sakamoto, "Inverse Source Problems for Diffusion Equations,", Ph.D. Thesis, (2010). [23] 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. doi: 10.1016/j.jmaa.2011.04.058.

show all references

##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1975). [2] M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition,, Inverse Problems, 13 (1997), 19. doi: 10.1088/0266-5611/13/1/003. [3] M. Choulli and M. Yamamoto, Generic well-posedness of an inverse parabolic problem--the Hölder-space approach,, Inverse Problems, 12 (1996), 195. doi: 10.1088/0266-5611/12/3/002. [4] M. Choulli and M. Yamamoto, Generic well-posedness of a linear inverse parabolic problem with diffusion parameters,, J. Inverse Ill-Posed Problems, 7 (1999), 241. doi: 10.1515/jiip.1999.7.3.241. [5] M. Ginoa, S. Gerbelli and H. E. Roman, Fractional diffusion equation and relaxation in complex viscoelastic materials,, Physica A, 191 (1992), 449. doi: 10.1016/0378-4371(92)90566-9. [6] R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, (1997), 223. [7] D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lec. Notes in Math. 840, (1981). [8] K. H. Hoffmann and M. Yamamoto, Generic uniqueness and stability in some inverse parabolic problem,, in, 422 (1993), 49. [9] V. Isakov, Inverse parabolic problems with the final overdetermination,, Comm. Pure Appl. Math., 44 (1991), 185. doi: 10.1002/cpa.3160440203. [10] F. John, "Partial Differential Equations,", Springer-Verlag, (1982). [11] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1976). [12] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations,", Elsevier, (2006). [13] F. Mainardi, On the initial value problem for the fractional diffusion-wave equation,, Waves and Stability in Continuous Media, (1994), 246. [14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation,, Appl. Math. Lett., 9 (1996), 23. doi: 10.1016/0893-9659(96)00089-4. [15] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics,, Fractals and Fractional Calculus in Continuum Mechanics. (Edited by A. Carpinteri, (1997), 291. [16] R. R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry,, Phys. Stat. Sol. B, 133 (1986), 425. doi: 10.1002/pssb.2221330150. [17] K. B. Oldham and J. Spanier, "The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order,", Academic Press, (1974). [18] I. Podlubny, "Fractional Differential Equations,", Academic Press, (1999). [19] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, "Methods for Solving Inverse Problems in Mathematical Physics,", Marcel Dekker, (2000). [20] J. Prüss, "Evolutionary Integral Equations and Applications,", Birkhäuser, (1993). [21] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation,, J. Phys. A, 27 (1994), 3407. doi: 10.1088/0305-4470/27/10/017. [22] K. Sakamoto, "Inverse Source Problems for Diffusion Equations,", Ph.D. Thesis, (2010). [23] 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. doi: 10.1016/j.jmaa.2011.04.058.
 [1] Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014 [2] Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 [3] Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007 [4] Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations & Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007 [5] Caochuan Ma, Zaihong Jiang, Renhui Wan. Local well-posedness for the tropical climate model with fractional velocity diffusion. Kinetic & Related Models, 2016, 9 (3) : 551-570. doi: 10.3934/krm.2016006 [6] Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605 [7] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [8] Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097 [9] Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435 [10] Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781 [11] Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387 [12] I. Baldomá, Tere M. Seara. The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 323-347. doi: 10.3934/dcdsb.2008.10.323 [13] Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013 [14] Young-Sam Kwon. On the well-posedness of entropy solutions for conservation laws with source terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 933-949. doi: 10.3934/dcds.2009.25.933 [15] Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203 [16] Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033 [17] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [18] Nobu Kishimoto. Local well-posedness for the Cauchy problem of the quadratic Schrödinger equation with nonlinearity $\bar u^2$. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1123-1143. doi: 10.3934/cpaa.2008.7.1123 [19] Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007 [20] Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241

2017 Impact Factor: 0.631