doi: 10.3934/ipi.2022030
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Source identification problems for abstract semilinear nonlocal differential equations

1. 

Department of Mathematics, Hanoi National University of Education, No. 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam

2. 

Institute of Mathematics, Vietnam Academy of Science and Technology, No. 18 Hoang Quoc Viet, Hanoi, Vietnam

3. 

Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh, Vietnam

*Corresponding author: Nguyen Thi Van Anh

Received  September 2021 Revised  May 2022 Early access June 2022

Fund Project: This research is funded by the Simons Foundation Grant Targeted for Institute of Mathematics, Vietnam Academy of Science and Technology

In this paper, we investigate a source identification problem for a class of abstract nonlocal differential equations in separable Hilbert spaces. The existence of mild solutions and strong solutions for the problem of identifying parameter are obtained. Furthermore, we study the continuous dependence on the data and the regularity of the mild solutions and strong solutions of nonlocal differential equations. Examples given in anomalous diffusion equations illustrate the existence and regularity results.

Citation: Nguyen Thi Van Anh, Bui Thi Hai Yen. Source identification problems for abstract semilinear nonlocal differential equations. Inverse Problems and Imaging, doi: 10.3934/ipi.2022030
References:
[1]

A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002.

[2]

N. H. Can, Y. Zhou, N. H. Tuan and T. N. Thach, Regularized solution approximation of a fractional pseudo-parabolic problem with a nonlinear source term and random data, Chaos Solitons Fractals, 136 (2020), 109847, 14 pp. doi: 10.1016/j.chaos.2020.109847.

[3]

X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.

[4]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

[5]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

[6]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[7]

D. N. Hào, J. Liu, N. V. Duc and N. V. Thang, Stability results for backward time-fractional parabolic equations, Inverse Problems, 35 (2019), 125006, 25 pp. doi: 10.1088/1361-6420/ab45d3.

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D. N. HàoH.-J. Reinhardt and A. Schneider, Stable approximation of fractional derivatives of rough functions, BIT, 35 (1995), 488-503.  doi: 10.1007/BF01739822.

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D. N. HàoN. V. DucN. V. Thang and N. T. Thanh, Regularization of backward time-fractional parabolic equations by Sobolev-type equations, J. Inverse Ill-Posed Probl., 28 (2020), 659-676.  doi: 10.1515/jiip-2020-0062.

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R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, World Science Publishing, River Edge, NJ, 2000, pp. 87–130. doi: 10.1142/9789812817747_0002.

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U. Hornung and R. E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl., 147 (1990), 69-80.  doi: 10.1016/0022-247X(90)90385-S.

[12]

J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data, Fract. Calc. Appl. Anal., 23 (2020), 1678-1701.  doi: 10.1515/fca-2020-0083.

[13]

J. Janno and K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Probl., 25 (2017), 777-798.  doi: 10.1515/jiip-2016-0082.

[14]

J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0.

[15]

J. Janno and A. Lorenzi, A parabolic integro-differential identification problem in a barrelled smooth domain, Z. Anal. Anwend., 25 (2006), 103-130.  doi: 10.4171/ZAA/1280.

[16]

T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces, J. Math. Anal. Appl., 483 (2020), 123655, 23 pp. doi: 10.1016/j.jmaa.2019.123655.

[17]

T. D. Ke and L. T. P. Thuy, Dissipativity and stability for semilinear anomalous diffusion equations involving delays, Math. Methods Appl. Sci., 43 (2020), 8449-8465.  doi: 10.1002/mma.6497.

[18]

T. D. Ke and T. V. Tuan, An identification problem involving fractional differential variational inequalities, J. Inverse Ill-Posed Probl., 29 (2021), 185-202.  doi: 10.1515/jiip-2017-0103.

[19]

J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb R^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.

[20]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations., 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.

[21]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.

[22]

Y.-H. Lin and H. Liu, Inverse problems for fractional equations with a minimal number of measurements, preprint, 2020, arXiv: 2203.03010.

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[24]

T. B. NgocN. H. Tuan and M. Kirane, Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source, J. Inverse Ill-Posed Probl., 28 (2020), 211-235.  doi: 10.1515/jiip-2018-0040.

[25]

T. B. Ngoc, N. H. Tuan and D. O'. Regan, Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104882, 13 pp. doi: 10.1016/j.cnsns.2019.104882.

[26]

J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666.  doi: 10.3934/dcds.2019026.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, 1$^st$ edition, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

N. N. Thang, Notes on ultraslow nonlocal telegraph evolution equations, Proc. Amer. Math. Soc.

[29]

D.-K. Tran and T.-P.-T. Lam, Nonlocal final value problem governed by semilinear anomalous diffusion equations, Evol. Equ. Control Theory, 9 (2020), 891-914.  doi: 10.3934/eect.2020038.

[30]

D.-K. Tran and N.-T. Nguyen, On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations, Commun. Pure Appl. Anal., 21 (2022), 817-835.  doi: 10.3934/cpaa.2021200.

[31]

N. H. TuanN. H. TuanD. Baleanu and T. N. Thach, On a backward problem for fractional diffusion equation with Riemann-Liouville derivative, Math. Methods Appl. Sci., 43 (2020), 1292-1312.  doi: 10.1002/mma.5943.

[32]

N. H. Tuan, Y. Zhou, L. D. Long and N. H. Can, Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative, Comput. Appl. Math., 39 (2020), Paper No. 75, 16 pp. doi: 10.1007/s40314-020-1103-2.

[33]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[34]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

show all references

References:
[1]

A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), 1176-1180.  doi: 10.1016/j.aml.2011.02.002.

[2]

N. H. Can, Y. Zhou, N. H. Tuan and T. N. Thach, Regularized solution approximation of a fractional pseudo-parabolic problem with a nonlinear source term and random data, Chaos Solitons Fractals, 136 (2020), 109847, 14 pp. doi: 10.1016/j.chaos.2020.109847.

[3]

X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17 (2019), 1861-1876.  doi: 10.4310/CMS.2019.v17.n7.a5.

[4]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

[5]

P. Clément and J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal., 12 (1981), 514-535.  doi: 10.1137/0512045.

[6]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[7]

D. N. Hào, J. Liu, N. V. Duc and N. V. Thang, Stability results for backward time-fractional parabolic equations, Inverse Problems, 35 (2019), 125006, 25 pp. doi: 10.1088/1361-6420/ab45d3.

[8]

D. N. HàoH.-J. Reinhardt and A. Schneider, Stable approximation of fractional derivatives of rough functions, BIT, 35 (1995), 488-503.  doi: 10.1007/BF01739822.

[9]

D. N. HàoN. V. DucN. V. Thang and N. T. Thanh, Regularization of backward time-fractional parabolic equations by Sobolev-type equations, J. Inverse Ill-Posed Probl., 28 (2020), 659-676.  doi: 10.1515/jiip-2020-0062.

[10]

R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, World Science Publishing, River Edge, NJ, 2000, pp. 87–130. doi: 10.1142/9789812817747_0002.

[11]

U. Hornung and R. E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl., 147 (1990), 69-80.  doi: 10.1016/0022-247X(90)90385-S.

[12]

J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data, Fract. Calc. Appl. Anal., 23 (2020), 1678-1701.  doi: 10.1515/fca-2020-0083.

[13]

J. Janno and K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, J. Inverse Ill-Posed Probl., 25 (2017), 777-798.  doi: 10.1515/jiip-2016-0082.

[14]

J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007, 19 pp. doi: 10.1088/1361-6420/aaa0f0.

[15]

J. Janno and A. Lorenzi, A parabolic integro-differential identification problem in a barrelled smooth domain, Z. Anal. Anwend., 25 (2006), 103-130.  doi: 10.4171/ZAA/1280.

[16]

T. D. Ke, N. N. Thang and L. T. P. Thuy, Regularity and stability analysis for a class of semilinear nonlocal differential equations in Hilbert spaces, J. Math. Anal. Appl., 483 (2020), 123655, 23 pp. doi: 10.1016/j.jmaa.2019.123655.

[17]

T. D. Ke and L. T. P. Thuy, Dissipativity and stability for semilinear anomalous diffusion equations involving delays, Math. Methods Appl. Sci., 43 (2020), 8449-8465.  doi: 10.1002/mma.6497.

[18]

T. D. Ke and T. V. Tuan, An identification problem involving fractional differential variational inequalities, J. Inverse Ill-Posed Probl., 29 (2021), 185-202.  doi: 10.1515/jiip-2017-0103.

[19]

J. KemppainenJ. SiljanderV. Vergara and R. Zacher, Decay estimates for time-fractional and other non-local in time subdiffusion equations in $\mathbb R^d$, Math. Ann., 366 (2016), 941-979.  doi: 10.1007/s00208-015-1356-z.

[20]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differential Equations., 263 (2017), 149-201.  doi: 10.1016/j.jde.2017.02.030.

[21]

A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024.

[22]

Y.-H. Lin and H. Liu, Inverse problems for fractional equations with a minimal number of measurements, preprint, 2020, arXiv: 2203.03010.

[23]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[24]

T. B. NgocN. H. Tuan and M. Kirane, Regularization of a sideways problem for a time-fractional diffusion equation with nonlinear source, J. Inverse Ill-Posed Probl., 28 (2020), 211-235.  doi: 10.1515/jiip-2018-0040.

[25]

T. B. Ngoc, N. H. Tuan and D. O'. Regan, Existence and uniqueness of mild solutions for a final value problem for nonlinear fractional diffusion systems, Commun. Nonlinear Sci. Numer. Simul., 78 (2019), 104882, 13 pp. doi: 10.1016/j.cnsns.2019.104882.

[26]

J. C. Pozo and V. Vergara, Fundamental solutions and decay of fully non-local problems, Discrete Contin. Dyn. Syst., 39 (2019), 639-666.  doi: 10.3934/dcds.2019026.

[27]

J. Prüss, Evolutionary Integral Equations and Applications, 1$^st$ edition, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[28]

N. N. Thang, Notes on ultraslow nonlocal telegraph evolution equations, Proc. Amer. Math. Soc.

[29]

D.-K. Tran and T.-P.-T. Lam, Nonlocal final value problem governed by semilinear anomalous diffusion equations, Evol. Equ. Control Theory, 9 (2020), 891-914.  doi: 10.3934/eect.2020038.

[30]

D.-K. Tran and N.-T. Nguyen, On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations, Commun. Pure Appl. Anal., 21 (2022), 817-835.  doi: 10.3934/cpaa.2021200.

[31]

N. H. TuanN. H. TuanD. Baleanu and T. N. Thach, On a backward problem for fractional diffusion equation with Riemann-Liouville derivative, Math. Methods Appl. Sci., 43 (2020), 1292-1312.  doi: 10.1002/mma.5943.

[32]

N. H. Tuan, Y. Zhou, L. D. Long and N. H. Can, Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative, Comput. Appl. Math., 39 (2020), Paper No. 75, 16 pp. doi: 10.1007/s40314-020-1103-2.

[33]

V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900.

[34]

V. Vergara and R. Zacher, Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations, J. Evol. Equ., 17 (2017), 599-626.  doi: 10.1007/s00028-016-0370-2.

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