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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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

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

[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.

[13]

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

[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.

[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. 

[16]

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

[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.

[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.

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

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

[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.

[25]

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

[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.

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

[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.

[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

[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.

[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.

[33]

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

[34]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

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

[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.

[13]

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

[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.

[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. 

[16]

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

[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.

[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.

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

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

[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.

[25]

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

[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.

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

[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.

[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

[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.

[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.

[33]

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

[34]

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

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