# American Institute of Mathematical Sciences

June  2016, 6(2): 251-269. doi: 10.3934/mcrf.2016003

## Determination of time dependent factors of coefficients in fractional diffusion equations

 1 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153, Japan 2 Aix-Marseille Université, CNRS, CPT UMR 7332, Marseille, 13288, France

Received  January 2015 Revised  February 2016 Published  April 2016

In the present paper, we consider initial-boundary value problems for partial differential equations with time-fractional derivatives which evolve in $Q=\Omega\times(0,T)$ where $\Omega$ is a bounded domain of $\mathbb{R}^d$ and $T>0$. We study the stability of the inverse problems of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at a point $x_0\in\overline{\Omega}$ for all $t\in(0,T)$.
Citation: Kenichi Fujishiro, Yavar Kian. Determination of time dependent factors of coefficients in fractional diffusion equations. Mathematical Control and Related Fields, 2016, 6 (2) : 251-269. doi: 10.3934/mcrf.2016003
##### 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. doi: 10.1029/92WR01757. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] O. P. Agarwal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn., 29 (2002), 145-155. doi: 10.1023/A:1016539022492. [4] S. Beckers and M. Yamamoto, Regularity and uniqueness of solution to linear diffusion equation with multiple time-fractional derivatives, International Series of Numerical Mathematics, 164 (2013), 45-55. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [6] A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. [7] J. R. Cannon and S. P. Esteva, An inverse problem for the heat equation, Inverse Problems, 2 (1986), 395-403. doi: 10.1088/0266-5611/2/4/007. [8] J. Carcione, F. Sanchez-Sesma, F. Luzón and J. Perez Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 345501, 23pp. doi: 10.1088/1751-8113/46/34/345501. [9] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002. [10] M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143. [11] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proceedings of the Japan Academy, 43 (1967), 82-86. doi: 10.3792/pja/1195521686. [12] P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. doi: 10.1088/0266-5611/29/6/065006. [13] V. D. Gejji and H. Jafari, Boundary value problems for fractional diffusion-wave equation, Aust. J. Math. Anal. Appl., 3 (2006), 1-8. [14] R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distributions and continuous time random walk, in Processes with long range correlations (eds. G. Rangarajan and M. Ding), Vol. 621, Lecture Notes in Physics. Berlin: Springer, (2003), 148-166. doi: 10.1007/3-540-44832-2_8. [15] 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. doi: 10.1029/98WR00214. [16] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind. 5A, 5A (2013), 51-57. [17] D. Henry, Geometric Theory of Semilinear Differential Equations, Springer-Verlag, Berlin, 1981. [18] Z. Li, O. Yu. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004. doi: 10.1088/0266-5611/32/1/015004. [19] J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. I, Dunod, Paris, 1968. [20] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem,, , (). [21] Y. Luchko, Initial-boundary value problems for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 374 (2011), 538-548. doi: 10.1016/j.jmaa.2010.08.048. [22] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. [23] D. Matignon, Stability properties for generalized fractional differential systems, ESAIM:Proc., 5 (1998), 145-158. doi: 10.1051/proc:1998004. [24] D. Matignon, An introduction to fractional calculus, in Scaling, Fractals and Wavelets, in Digital Signal and Image Processing Series (eds. P. Abry, P. Goncalvès and J. Lévy-Véhel), ISTE - Wiley, 7 (2009), 237-278. [25] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. [26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993. [27] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8pp. doi: 10.1088/0266-5611/29/7/075013. [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [29] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [30] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017. [31] S. Saitoh, V. K. Tuan and M. Yamamoto, Convolution inequalities and applications, J. Ineq. Pure and Appl. Math., 4 (2003), Art. 50, 8pp. [32] S. Saitoh, V. K. Tuan and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. Ineq. Pure and Appl. Math., 3 (2002), Art. 80, 11pp. [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), 426-447. doi: 10.1016/j.jmaa.2011.04.058. [34] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Philadelphia, 1993. [35] E. M. Stein, Singular Intearals and Differentiability Properties of Functions, Princeton university press, Princeton, 1970. [36] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [37] X. Xu, J. 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. [38] 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), 105010, 10pp. doi: 10.1088/0266-5611/28/10/105010.

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##### 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. doi: 10.1029/92WR01757. [2] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [3] O. P. Agarwal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dyn., 29 (2002), 145-155. doi: 10.1023/A:1016539022492. [4] S. Beckers and M. Yamamoto, Regularity and uniqueness of solution to linear diffusion equation with multiple time-fractional derivatives, International Series of Numerical Mathematics, 164 (2013), 45-55. [5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [6] A. L. Bukhgeim and M. V. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. [7] J. R. Cannon and S. P. Esteva, An inverse problem for the heat equation, Inverse Problems, 2 (1986), 395-403. doi: 10.1088/0266-5611/2/4/007. [8] J. Carcione, F. Sanchez-Sesma, F. Luzón and J. Perez Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 345501, 23pp. doi: 10.1088/1751-8113/46/34/345501. [9] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002, 16pp. doi: 10.1088/0266-5611/25/11/115002. [10] M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160. doi: 10.3934/mcrf.2013.3.143. [11] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proceedings of the Japan Academy, 43 (1967), 82-86. doi: 10.3792/pja/1195521686. [12] P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. doi: 10.1088/0266-5611/29/6/065006. [13] V. D. Gejji and H. Jafari, Boundary value problems for fractional diffusion-wave equation, Aust. J. Math. Anal. Appl., 3 (2006), 1-8. [14] R. Gorenflo and F. Mainardi, Fractional diffusion processes: Probability distributions and continuous time random walk, in Processes with long range correlations (eds. G. Rangarajan and M. Ding), Vol. 621, Lecture Notes in Physics. Berlin: Springer, (2003), 148-166. doi: 10.1007/3-540-44832-2_8. [15] 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. doi: 10.1029/98WR00214. [16] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind. 5A, 5A (2013), 51-57. [17] D. Henry, Geometric Theory of Semilinear Differential Equations, Springer-Verlag, Berlin, 1981. [18] Z. Li, O. Yu. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004. doi: 10.1088/0266-5611/32/1/015004. [19] J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. I, Dunod, Paris, 1968. [20] Y. Liu, W. Rundell and M. Yamamoto, Strong maximum principle for fractional diffusion equations and an application to an inverse source problem,, , (). [21] Y. Luchko, Initial-boundary value problems for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 374 (2011), 538-548. doi: 10.1016/j.jmaa.2010.08.048. [22] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018. [23] D. Matignon, Stability properties for generalized fractional differential systems, ESAIM:Proc., 5 (1998), 145-158. doi: 10.1051/proc:1998004. [24] D. Matignon, An introduction to fractional calculus, in Scaling, Fractals and Wavelets, in Digital Signal and Image Processing Series (eds. P. Abry, P. Goncalvès and J. Lévy-Véhel), ISTE - Wiley, 7 (2009), 237-278. [25] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics reports, 339 (2000), 1-77. doi: 10.1016/S0370-1573(00)00070-3. [26] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993. [27] L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems, 29 (2013), 075013, 8pp. doi: 10.1088/0266-5611/29/7/075013. [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [29] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [30] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A, 27 (1994), 3407-3410. doi: 10.1088/0305-4470/27/10/017. [31] S. Saitoh, V. K. Tuan and M. Yamamoto, Convolution inequalities and applications, J. Ineq. Pure and Appl. Math., 4 (2003), Art. 50, 8pp. [32] S. Saitoh, V. K. Tuan and M. Yamamoto, Reverse convolution inequalities and applications to inverse heat source problems, J. Ineq. Pure and Appl. Math., 3 (2002), Art. 80, 11pp. [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), 426-447. doi: 10.1016/j.jmaa.2011.04.058. [34] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Philadelphia, 1993. [35] E. M. Stein, Singular Intearals and Differentiability Properties of Functions, Princeton university press, Princeton, 1970. [36] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech., 16 (1967), 1031-1060. [37] X. Xu, J. 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. [38] 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), 105010, 10pp. doi: 10.1088/0266-5611/28/10/105010.
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