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February  2017, 11(1): 125-149. doi: 10.3934/ipi.2017007

Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation

Jaan Janno and  Kairi Kasemets

Tallinn University of Technology, Ehitajate tee 5, Tallinn 19086, Estonia

Received  November 2015 Revised  June 2016 Published  January 2017

Fund Project: The research is supported by the Estonian Research Council grant PUT568 and institutional research funding IUT33-24 of the Estonian Ministry of Education and Research.

An inverse problem to determine a space-dependent factor in a semilinear time-fractional diffusion equation is considered. Additional data are given in the form of an integral with the Borel measure over the time. Uniqueness of the solution of the inverse problem is studied. The method uses a positivity principle of the corresponding differential equation that is also proved in the paper.

Keywords: Inverse problem, fractional diffusion, semilinear fractional parabolic equation, uniqueness, positivity principle.
Mathematics Subject Classification: Primary: 35R30; Secondary: 80A23.
Citation: 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
References:
[1]

M. Al-Refai and Y. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives, Appl. Math. Comput., 257 (2015), 40-51.  doi: 10.1016/j.amc.2014.12.127.  Google Scholar

[2]

E. Beretta and C. Cavaterra, Identifying a space-dependent coefficient in a reaction-diffusion equation, Inverse Problems and Imaging, 5 (2011), 285-296.  doi: 10.3934/ipi.2011.5.285.  Google Scholar

[3]

H. Brunner, H. Han and D. Yin, The maximum principle for time-fractional diffusion equations and its application, Numer. Funct. Anal Optim., 36 (2015), 1307-1321.  doi: 10.1080/01630563.2015.1065887.  Google Scholar

[4]

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.  Google Scholar

[5]

K. M. Furati, O. S. Iyiola and M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. Comput., 249 (2014), 24-31.  doi: 10.1016/j.amc.2014.10.046.  Google Scholar

[6]

V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math., 220 (2008), 215-225.  doi: 10.1016/j.cam.2007.08.011.  Google Scholar

[7]

R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes, J. Comput. Appl. Math., 229 (2009), 400-415.  doi: 10.1016/j.cam.2008.04.005.  Google Scholar

[8]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, Math. Zeitschrift, 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[9]

V. Isakov, Inverse parabolic problems with final overdetermination, Commun. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.  Google Scholar

[10] V. Isakov, Inverse Problems for Partial Differential Equations, 2 edition, Springer, New York, 2006.   Google Scholar
[11]

J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Diff. Eqns. 2016 (2016), 28pp.  Google Scholar

[12]

J. Janno and K. Kasemets, A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination, Inverse Problems and Imaging, 3 (2009), 17-41.  doi: 10.3934/ipi.2009.3.17.  Google Scholar

[13]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28 (2012), 075010, 19pp.  Google Scholar

[14]

M. Kirane, A. S. Malik and M. A. Al-Gwaizb, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci., 36 (2013), 1056-1069.  doi: 10.1002/mma.2661.  Google Scholar

[15]

M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlin. Stud., 20 (2013), 591-621.   Google Scholar

[16]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type AMS, Providence, Rhode Island, 1968. Google Scholar

[17]

Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048.  Google Scholar

[18]

Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems 29 (2013), 065019, 16pp.  Google Scholar

[19] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.   Google Scholar
[20]

R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Computers Math. Appl., 59 (2010), 1586-1593.  doi: 10.1016/j.camwa.2009.08.039.  Google Scholar

[21] C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970.   Google Scholar
[22] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Berlin, 1993.   Google Scholar
[23]

S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 3847-3854.  doi: 10.1016/j.cnsns.2010.02.007.  Google Scholar

[24]

K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1 (2011), 509-518.  doi: 10.3934/mcrf.2011.1.509.  Google Scholar

[25]

K. Seki, M. Wojcik and M. Tachiya, Fractional reaction-diffusion equation, J. Chem. Phys., 119 (2003), 2165-2170.  doi: 10.1063/1.1587126.  Google Scholar

[26]

H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259 (1980), 299-310.  doi: 10.1090/S0002-9947-1980-0561838-5.  Google Scholar

[27] V. E. Tarasov, Fractional Dynamics. Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2010.   Google Scholar
[28]

V. Turut and N. Güzel, Comparing numerical methods for solving time-fractional reaction-diffusion equations, Intern. Scholar. Res. Notices 2012 (2012), Art. ID 737206, 28 pp.  Google Scholar

[29]

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.  doi: 10.1016/j.apnum.2013.12.002.  Google Scholar

[30]

R. Zacher, Quasilinear Parabolic Problems with Nonlinear Boundary Conditions Ph. D thesis, Martin-Luther-Universität Halle-Wittenberg, 2003. Available from: https://www.yumpu.com/en/document/view/4926858/quasilinear-parabolic-problems-with-nonlinear-boundary-conditions Google Scholar

[31]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[32]

R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.  doi: 10.1016/j.jmaa.2008.06.054.  Google Scholar

[33]

G. M. Zaslavsky, Fractional kinetics and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.  Google Scholar

[34]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems 27 (2011), 035010, 12pp.  Google Scholar

show all references

References:
[1]

M. Al-Refai and Y. Luchko, Maximum principle for the multi-term time-fractional diffusion equations with the Riemann-Liouville fractional derivatives, Appl. Math. Comput., 257 (2015), 40-51.  doi: 10.1016/j.amc.2014.12.127.  Google Scholar

[2]

E. Beretta and C. Cavaterra, Identifying a space-dependent coefficient in a reaction-diffusion equation, Inverse Problems and Imaging, 5 (2011), 285-296.  doi: 10.3934/ipi.2011.5.285.  Google Scholar

[3]

H. Brunner, H. Han and D. Yin, The maximum principle for time-fractional diffusion equations and its application, Numer. Funct. Anal Optim., 36 (2015), 1307-1321.  doi: 10.1080/01630563.2015.1065887.  Google Scholar

[4]

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.  Google Scholar

[5]

K. M. Furati, O. S. Iyiola and M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. Comput., 249 (2014), 24-31.  doi: 10.1016/j.amc.2014.10.046.  Google Scholar

[6]

V. Gafiychuk, B. Datsko and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Comput. Appl. Math., 220 (2008), 215-225.  doi: 10.1016/j.cam.2007.08.011.  Google Scholar

[7]

R. Gorenflo and F. Mainardi, Some recent advances in theory and simulation of fractional diffusion processes, J. Comput. Appl. Math., 229 (2009), 400-415.  doi: 10.1016/j.cam.2008.04.005.  Google Scholar

[8]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals, Math. Zeitschrift, 27 (1928), 565-606.  doi: 10.1007/BF01171116.  Google Scholar

[9]

V. Isakov, Inverse parabolic problems with final overdetermination, Commun. Pure Appl. Math., 44 (1991), 185-209.  doi: 10.1002/cpa.3160440203.  Google Scholar

[10] V. Isakov, Inverse Problems for Partial Differential Equations, 2 edition, Springer, New York, 2006.   Google Scholar
[11]

J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Diff. Eqns. 2016 (2016), 28pp.  Google Scholar

[12]

J. Janno and K. Kasemets, A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination, Inverse Problems and Imaging, 3 (2009), 17-41.  doi: 10.3934/ipi.2009.3.17.  Google Scholar

[13]

B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28 (2012), 075010, 19pp.  Google Scholar

[14]

M. Kirane, A. S. Malik and M. A. Al-Gwaizb, An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions, Math. Meth. Appl. Sci., 36 (2013), 1056-1069.  doi: 10.1002/mma.2661.  Google Scholar

[15]

M. Krasnoschok and N. Vasylyeva, On a solvability of a nonlinear fractional reaction-diffusion system in the Hölder spaces, Nonlin. Stud., 20 (2013), 591-621.   Google Scholar

[16]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type AMS, Providence, Rhode Island, 1968. Google Scholar

[17]

Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), 538-548.  doi: 10.1016/j.jmaa.2010.08.048.  Google Scholar

[18]

Y. Luchko, W. Rundell, M. Yamamoto and L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Problems 29 (2013), 065019, 16pp.  Google Scholar

[19] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995.   Google Scholar
[20]

R. L. Magin, Fractional calculus models of complex dynamics in biological tissues, Computers Math. Appl., 59 (2010), 1586-1593.  doi: 10.1016/j.camwa.2009.08.039.  Google Scholar

[21] C. Miranda, Partial Differential Equations of Elliptic Type, Springer-Verlag, New York, 1970.   Google Scholar
[22] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Berlin, 1993.   Google Scholar
[23]

S. Z. Rida, A. M. A. El-Sayed and A. A. M. Arafa, On the solutions of time-fractional reaction-diffusion equations, Commun. Nonlinear Sci. Numer. Simulat., 15 (2010), 3847-3854.  doi: 10.1016/j.cnsns.2010.02.007.  Google Scholar

[24]

K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1 (2011), 509-518.  doi: 10.3934/mcrf.2011.1.509.  Google Scholar

[25]

K. Seki, M. Wojcik and M. Tachiya, Fractional reaction-diffusion equation, J. Chem. Phys., 119 (2003), 2165-2170.  doi: 10.1063/1.1587126.  Google Scholar

[26]

H. B. Stewart, Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 259 (1980), 299-310.  doi: 10.1090/S0002-9947-1980-0561838-5.  Google Scholar

[27] V. E. Tarasov, Fractional Dynamics. Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, New York, 2010.   Google Scholar
[28]

V. Turut and N. Güzel, Comparing numerical methods for solving time-fractional reaction-diffusion equations, Intern. Scholar. Res. Notices 2012 (2012), Art. ID 737206, 28 pp.  Google Scholar

[29]

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.  doi: 10.1016/j.apnum.2013.12.002.  Google Scholar

[30]

R. Zacher, Quasilinear Parabolic Problems with Nonlinear Boundary Conditions Ph. D thesis, Martin-Luther-Universität Halle-Wittenberg, 2003. Available from: https://www.yumpu.com/en/document/view/4926858/quasilinear-parabolic-problems-with-nonlinear-boundary-conditions Google Scholar

[31]

R. Zacher, Maximal regularity of type $L_p$ for abstract parabolic Volterra equations, J. Evol. Equ., 5 (2005), 79-103.  doi: 10.1007/s00028-004-0161-z.  Google Scholar

[32]

R. Zacher, Boundedness of weak solutions to evolutionary partial integro-differential equations with discontinuous coefficients, J. Math. Anal. Appl., 348 (2008), 137-149.  doi: 10.1016/j.jmaa.2008.06.054.  Google Scholar

[33]

G. M. Zaslavsky, Fractional kinetics and anomalous transport, Physics Reports, 371 (2002), 461-580.  doi: 10.1016/S0370-1573(02)00331-9.  Google Scholar

[34]

Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Problems 27 (2011), 035010, 12pp.  Google Scholar

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