Source identification problems for a class of subdiffusion equations with weak nonlinearities

Nguyen Thi Van Anh; Nguyen Nhu Quan

doi:10.3934/mcrf.2024024

Article Contents

2025, Volume 15, Issue 2: 548-569. Doi: 10.3934/mcrf.2024024

This issue Previous Article Strong and exponential stabilization of linear boundary control systems Next Article Local Halanay's inequality with application to feedback stabilization

Source identification problems for a class of subdiffusion equations with weak nonlinearities

Nguyen Thi Van Anh^{1, 2,} and
Nguyen Nhu Quan^3, ,

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, Electric Power University, No. 235 Hoang Quoc Viet, Bac Tu Liem, Hanoi, Vietnam

^*Corresponding author: Nguyen Nhu Quan
^*Corresponding author: Nguyen Nhu Quan

Received: January 2024

Revised: May 2024

Early access: June 2024

Published: June 2025

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Abstract

The paper deals with a source identification problem of the subdiffusion equations from the initial value and nonlocal final data observations where the nonlinearity likely takes values in Hilbert scales spaces. The existence and uniqueness results are proved by establishing some estimates for resolvent operators using the embedding theorem and some fixed point arguments. We also study the regularity results for these equations, more especifically we prove the Hölder continuity of mild solutions.

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Mathematics Subject Classification: 34G20, 34K29, 35R11, 47N20, 93B30.

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[1]	F. Amblard, A. C. Maggs, B. Yurke, A. N. Pargellis, S. Leibler, Y. Benoist, P. Foulon and F. Labourie, Subdiffusion and anomalous local viscoelasticity in actin networks, Phys. Rev. Lett., 77 (1996), 4470-4473.
[2]	N. T. V. Anh and B. T. H. Yen, Source identification problems for abstract semilinear nonlocal differential equations, Inverse Probl. Imaging, 16 (2022), 1389-1428. doi: 10.3934/ipi.2022030.
[3]	N. T. V. Anh and B. T. H. Yen, On the time-delayed anomalous diffusion equations with nonlocal initial conditions, Communications on Pure and Applied Analysis, 21 (2022), 3701-3719. doi: 10.3934/cpaa.2022119.
[4]	N. T. V. Anh, T. D. Ke and D. Lan, The final value problem for anomalous diffusion equations involving weak-valued nonlinearities, Journal of Mathematical Analysis and Applications, 532 (2024), 127916, 20 pp. doi: 10.1016/j.jmaa.2023.127916.
[5]	N. T. Anh, T. D. Ke and N. N. Quan, Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3637-3654. doi: 10.3934/dcdsb.2016114.
[6]	M. Ali, S. Aziz and S. A. Malik, Inverse problem for a multi-term fractional differential equation, Fractional Calculus and Applied Analysis, 23 (2020), 799-821. doi: 10.1515/fca-2020-0040.
[7]	N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1989.
[8]	P. K. Bhattacharyya, Distributions: Generalized Functions with Applications in Sobolev Spaces, De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012. doi: 10.1515/9783110269291.
[9]	J. Crank, The Mathematics of Diffusion, 2$^{nd}$ edition, Oxford University Press, 1989.
[10]	N.V. Dac, T.D. Ke and L.T.P. Thuy, On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearities, Discrete and Continuous Dynamical Systems - S, 16 (2023), 2883-2901. doi: 10.3934/dcdss.2023071.
[11]	W. Feller, An Introduction to Probability Theory and Its Applications, 2$^{nd}$ edition, John Wiley & Sons, New York, 1971.
[12]	M. Fritz, C. Kuttler, M. L. Rajendran, B. Wohlmuth and L. Scarabosio, On a subdiffusive tumour growth model with fractional time derivative, IMA Journal of Applied Mathematics, 86 (2021), 688-729. doi: 10.1093/imamat/hxab009.
[13]	C. G. Gal and M. Warma, Fractional-in-Time Semilinear Parabolic Equations and Applications, Math. Appl. (Berlin), 84. Springer, Cham, 2020. doi: 10.1007/978-3-030-45043-4.
[14]	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.
[15]	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.
[16]	J. Janno and K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion, Inverse Ill-Posed Probl., 25 (2017), 777-798. doi: 10.1515/jiip-2016-0082.
[17]	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.
[18]	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.
[19]	T. D. Ke and L. T. P. Thuy, Nonlocal final value problem governed by semilinear anomalous diffusion equations, Evol. Equ. Control Theory, 9 (2020), 891-914. doi: 10.3934/eect.2020038.
[20]	T. D. Ke, L. T. P. Thuy and P. T. Tuan, An inverse source problem for generalized Rayleigh-Stokes equations involving superlinear perturbations, J. Math. Anal. Appl., 507 (2022), 125797, 24 pp. doi: 10.1016/j.jmaa.2021.125797.
[21]	J. Kemppainen, J. Siljander, V. 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.
[22]	J. T. Katsikadelis, Numerical solution of multi-term fractional differential equations, ZAMM Z. Angew. Math. Mech., 89.7 (2009), 593-608. doi: 10.1002/zamm.200900252.
[23]	A. Kubica, K. Ryszewska and M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, SpringerBriefs Math., Springer, Singapore, 2020. doi: 10.1007/978-981-15-9066-5.
[24]	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.
[25]	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.
[26]	D. N. Hào, N. V. Duc, N. 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.
[27]	C. Li, W. Deng and L. Zhao, Well-posedness and numerical algorithm for the tempered fractional differential equations, Discrete and Continuous Dynamical Systems-B, 24 (2019), 1989-2015. doi: 10.3934/dcdsb.2019026.
[28]	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.
[29]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Intersci. Publ., John Wiley & Sons, Inc., New York, 1993.
[30]	G. J. Morales, Two-dimensional chaotic thermostat and behavior of a thermalized charge in a weak magnetic field, Physical Review E, 99 (2019), 062218.
[31]	T. B. Ngoc, N. H. Tuan and D. 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.
[32]	L. I. Palade, P. Attané, R. R. Huilgol and B. Mena, Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models, Inter. J. Eng. Sci., 37 (1999), 315-329. doi: 10.1016/S0020-7225(98)00080-9.
[33]	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.
[34]	J. Prüss, Evolutionary Integral Equations and Applications, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.
[35]	S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London, 1993.
[36]	F. Sabzikar, M. M. Meerschaert and J. Chen, Tempered fractional calculus, Journal of Computational Physics, 293 (2015), 14-28. doi: 10.1016/j.jcp.2014.04.024.
[37]	D. M. Tartakovsky and M. Dentz, Diffusion in Porous Media: Phenomena and Mechanisms, Transp Porous Med., 130 (2019), 105-127. doi: 10.1007/s11242-019-01262-6.
[38]	N. H. Tuan, N. H. Tuan, D. 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.
[39]	T. V. Tuan, Stability and regularity in inverse source problem for generalized subdiffusion equation perturbed by locally Lipschitz sources, Z. Angew. Math. Phys., 74 (2023), Paper No. 65, 25 pp. doi: 10.1007/s00033-023-01958-2.
[40]	E. R. Weeks and H. L. Swinney, Anomalous diffusion resulting from strongly asymmetric random walks, Phys. Rev. E, 57 (1998), 4915-4920. doi: 10.1103/PhysRevE.57.4915.
[41]	L. Zhao, W. Deng and J. S. Hesthaven, Spectral methods for tempered fractional differential equations, Mathematics of Computation, (2016).