Abstract
Under study is the multidimensional inverse problem of determining the convolutional kernel of the integral term in an integro-differential wave equation. The direct problem is represented by a generalized initial-boundary value problem for this equation with zero initial data and the Neumann boundary condition in the form of the Dirac delta-function. For solving the inverse problem, the traces of the solution to the direct problem on the domain boundary are given as an additional condition. The main result of the article is the theorem of global unique solvability of the inverse problem in the class of functions continuous in the time variable \( t \) and analytic in the space variable. We apply the methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions.
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References
Lavrentev M. M., Romanov V. G., and Vasilev V. G., Multidimensional Inverse Problems for Differential Equations [Russian], Nauka, Novosibirsk (1969).
Romanov V. G., Stability in Inverse Problems [Russian], Nauchnyi Mir, Moscow (2005).
Lorenzi A., “Identification problems for integro-differential equations,” in: Ill-Posed Problems in Natural Sciences, Tikhonov A., Ed., TVP Sci. Publ., Moscow (1992), 342–366.
Bukhgeim A. L., “Inverse problems of memory reconstruction,” J. Inverse Ill Posed Probl., vol. 1, no. 3, 193–206 (1993).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equation,” Math. Meth. Appl. Sci., vol. 41, no. 17, 8019–8032 (2018).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the one-dimensional kernel of viscoelasticity equation with a source of explosive type,” J. Inverse Ill Posed Probl., vol. 28, no. 1, 43–52 (2020).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the one-dimensional kernel of the viscoelasticity equation,” Sib. Zh. Ind. Mat., vol. 16, no. 2, 72–82 (2013).
Durdiev D. K., “Global solvability of an inverse problem for an integro-differential equation of electrodynamics,” Differ. Equ., vol. 44, no. 7, 893–899 (2008).
Bukhgeim A. L. and Kalinina N. I., “Global convergence of the Newton method in the inverse problems of memory reconstruction,” Sib. Math. J., vol. 38, no. 5, 881–895 (1997).
Janno J. and von Wolfersdorf L., “Inverse problems for identification of memory kernels in viscoelasticity,” Math. Methods Appl. Sci., vol. 20, no. 4, 291–314 (1997).
Ovsyannikov L. V., “A singular operator in a scale of Banach spaces,” Soviet Math., Dokl., vol. 6, no. 4, 1025–1028 (1965).
Ovsyannikov L. V., “A nonlocal Cauchy problem in scales of Banach spaces,” Dokl. Akad. Nauk SSSR, vol. 200, no. 4, 789–792 (1971).
Ovsyannikov L. V., Analytic Groups [Russian], Nauka, Novosibirsk (1972).
Nirenberg L., Topics in Nonlinear Functional Analysis, Courant Inst. Math. Sci. and New York Univ., New York (1974).
Romanov V. G., “Local solvability of some multidimensional inverse problems for hyperbolic equations,” Differ. Equ., vol. 25, no. 2, 203–209 (1989).
Romanov V. G., “Questions of well-posedness of the problem of determining the speed of sound,” Sib. Mat. Zh., vol. 30, no. 4, 125–134 (1989).
Romanov V. G., “On solvability of inverse problems for hyperbolic equations in a class of functions analytic in part of variables,” Dokl. Akad. Nauk SSSR, vol. 304, no. 4, 807–811 (1989).
Bozorov Z. R., “The problem of determining the two-dimensional kernel of a viscoelasticity equation,” J. Appl. Ind. Math., vol. 23, no. 1, 20–36 (2020).
Durdiev D. K. and Rahmonov A. A., “The problem of determining the 2D kernel in a system of integro-differential equations of a viscoelastic porous medium,” J. Appl. Ind. Math., vol. 14, no. 2, 281–295 (2020).
Durdiev D. K. and Totieva Zh. D., “The problem of determining the multidimensional kernel of the viscoelasticity equation,” Vladikavkaz. Mat. Zh., vol. 17, no. 4, 18–43 (2015).
Durdiev D. K. and Safarov Zh. Sh., “Local solvability of the problem of definition of the spatial part of the multidimensional kernel in an integro-differential equation of hyperbolic type,” Vestnik Samarsk. Gos. Univ. Ser. Fiz.-Mat. Nauki, vol. 29, no. 4, 37–47 (2012).
Durdiev D. K., “Some multidimensional inverse problems of memory determination in hyperbolic equations,” Zh. Mat. Fiz., Anal. Geom., vol. 3, no. 4, 411–423 (2007).
Durdiev D. K., “A multidimensional inverse problem for an equation with memory,” Sib. Math. J., vol. 35, no. 3, 514–521 (1994).
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Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 2, pp. 269–285. https://doi.org/10.33048/smzh.2021.62.203
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Durdiev, D.K., Totieva, Z.D. About Global Solvability of a Multidimensional Inverse Problem for an Equation with Memory. Sib Math J 62, 215–229 (2021). https://doi.org/10.1134/S0037446621020038
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DOI: https://doi.org/10.1134/S0037446621020038