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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 12
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Research Article

The backward problem for time-fractional evolution equations

ORCID Icon, ORCID Icon &
Pages 2194-2212 | Received 02 Feb 2023, Accepted 23 Nov 2023, Published online: 04 Dec 2023

References

  • Agmon S, Nirenberg L. Properties of solutions of ordinary differential equations in Banach space. Comm Pure Appl Math. 1963;16(2):121–239. doi:10.1002/cpa.v16:2
  • Isakov V. Inverse problems for partial differential equations. Berlin: Springer; 2017.
  • Payne LE. Improperly posed problems in partial differential equations. Philadelphia: SIAM; 1975.
  • Adams EE, Gelhar LW. Field study of dispersion in a heterogeneous aquifer 2 spatial moments analysis. Water Resources Res. 1992;28(12):3293–3307. doi:10.1029/92WR01757
  • Oldham KB, Spanier J. The fractional calculus: theory and applications of differentiation and integration to arbitrary order. New York: Academic Press; 1974.
  • Ait Ben Hassi EM, Chorfi SE, Maniar L. An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions. J Inverse Ill-Posed Probl. 2022;30(3):363–378. doi:10.1515/jiip-2020-0067
  • Ait Ben Hassi EM, Chorfi SE, Maniar L. Inverse problems for general parabolic systems and application to Ornstein–Uhlenbeck equation. Discrete Cont Dyn Syst - S. 2023. doi: 10.3934/dcdss.2022212
  • Krein SG, Prozorovskaya OI. Analytic semigroups and incorrect problems for evolutionary equations. Dokl Akad Nauk SSSR. 1960;133:277–280.
  • Yamamoto M, Zou J. Simultaneous reconstruction of the initial temperature and heat radiative coefficient. Inverse Probl. 2001;17:1181–1202. doi:10.1088/0266-5611/17/4/340
  • Tuan NH, Caraballo T, Ngoc TB, et al. Existence and regularity results for terminal value problem for nonlinear fractional wave equations. Nonlinearity. 2021;34(3):1448–1502. doi:10.1088/1361-6544/abc4d9
  • Wei T, Zhang Y. The backward problem for a time-fractional diffusion-wave equation in a bounded domain. Comput Math with Appl. 2018;75(10):3632–3648. doi:10.1016/j.camwa.2018.02.022
  • Li J, Yamamoto M, Zou J. Conditional stability and numerical reconstruction of initial temperature. Commun Pure Appl Anal. 2009;8(1):361–382. doi:10.3934/cpaa.2009.8.361
  • Cheng J, Yamamoto M. One new strategy for a priori choice of regularizing parameters in Tikhonov's regularization. Inverse Probl. 2000;16(4):L31–L38. doi:10.1088/0266-5611/16/4/101
  • Martinez C, Sanz M. The Theory of Fractional Powers of Operators. Amsterdam: Elsevier Science; 2001. (North-Holland Mathematics Studies).
  • Sakamoto K, Yamamoto M. Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl. 2011;382(1):426–447. doi:10.1016/j.jmaa.2011.04.058
  • Floridia G, Li Z, Yamamoto M. Well-posedness for the backward problems in time for general time-fractional diffusion equation. Atti Accad Naz Lincei Cl Sci Fis Mat Natur. 2020;31(3):593–610. doi:10.4171/RLM
  • Niculescu CP, Persson L-E. Convex functions and their applications. Cham Switzerland: Springer; 2018.
  • Widder DV. The Laplace transforms. Princeton: Princeton University Press; 1946.
  • Schilling LR, Song R, Vondraček Z. Bernstein functions: theory and applications. Berlin, Boston: De Gruyter; 2012.
  • Podlubny I. Fractional differential equations. San Diego: Academic Press; 1999.
  • Gorenflo R, Mainardi F. Fractional calculus: Integral and differential equations of fractional order. In: Carpinteri A, Mainardi F, editors. Fractals and Fractional Calculus in Continuum Mechanics; New York: Springer-Verlag; 1997. p. 223–276.
  • Pollard H. The complete monotonic character of the Mittag–Leffler function Eα(−x). Bull Amer Math Soc. 1948;54(12):1115–1116. doi:10.1090/bull/1948-54-12
  • Schneider WR. Completely monotone generalized Mittag–Leffler functions. Expo Math. 1996;14:3–16.
  • Engel K-J, Nagel R. One-parameter semigroups for linear evolution equations. New York: Springer; 2000. (Graduate Texts in Mathematics; 194).
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. Berlin: Springer-Verlag; 1983.
  • Liu JJ, Yamamoto M. A backward problem for the time-fractional diffusion equation. Appl Anal. 2010;89(11):1769–1788. doi:10.1080/00036810903479731
  • Hào DN, Liu J, Van Duc N, et al. Stability results for backward time-fractional parabolic equations. Inverse Probl. 2019;35(12):125006. doi:10.1088/1361-6420/ab45d3
  • Campiti M, Metafune G, Pallara D. Degenerate self-adjoint evolution equations on the unit interval. Semigroup Forum. 1998;57(1):1–36. doi:10.1007/PL00005959
  • Cannarsa P, Martinez P, Vancostenoble J. Null controllability of degenerate heat equations. Adv Differ Equ. 2005;10:153–90.
  • Alabau-Boussouira F, Cannarsa P, Fragnelli G. Carleman estimates for degenerate parabolic operators with applications to null controllability. J Evol Equ. 2006;6(2):161–204. doi:10.1007/s00028-006-0222-6
  • Maniar L, Meyries M, Schnaubelt R. Null controllability for parabolic equations with dynamic boundary conditions. Evol Equ Control Theory. 2017;6(3):381–407. doi:10.3934/eect.2017020

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