The problem of determining the multidimensional kernel of viscoelasticity equation. (Russian. English summary) Zbl 1474.45100

Durdiev, Durdimurod Kalandarovich; Totieva, Zhanna Dmitrievna

Vladikavkaz. Mat. Zh. 17, No. 4, 18-43 (2015).

Summary: The integro-differential system of viscoelasticity equations is considered. The direct problem of determining of the displacements vector from the initial-boundary problem for this system is formulated. It is assumed that the kernel in the integral part depends on both the time and the space variable \(x_2\). For its determination an additional condition relative to the first component of the displacements vector with \(x_3=0\) is posed. The inverse problem is replaced by the equivalent system of integral equations. The study is based on the method of scales of Banach spaces of analytic functions. The theorem on local unique solvability of the inverse problem is proved in the class of functions analytic on the variable \(x_2\) and continuous on \(t\).

MSC:

45Q05	Inverse problems for integral equations
45K05	Integro-partial differential equations
74D05	Linear constitutive equations for materials with memory

Keywords:

inverse problem; stability; delta function; Lame’s coefficients; kernel

References:

[1]	Tuaeva Zh. D., “Mnogomernaya matematicheskaya model seismiki s pamyatyu”, Issledovaniya po dif. uravneniyam i mat. modelirovaniyu, Itogi nauki. YuFO. Mat. forum, 1, ch. 2, VNTs RAN, Vladikavkaz, 2008, 297-306
[2]	Durdiev D. K., Totieva Zh. D., “Zadacha ob opredelenii odnomernogo yadra uravneniya vyazkouprugosti”, Sib. zhurn. industr. matem., 16:2 (2013), 72-82 · Zbl 1340.35382
[3]	Ovsyannikov L. V., “Singulyarnyi operator v shkale banakhovykh prostranstv”, Dokl. AN SSSR, 163:4 (1965), 819-822 · Zbl 0144.39003
[4]	Ovsyannikov L. V., “Nelineinaya zadacha Koshi v shkalakh banakhovykh prostranstv”, Dokl. AN SSSR, 200:4 (1971), 789-792 · Zbl 0234.35018
[5]	Nirenberg L., Topics in Nonlinear Functional Analysis, Courant Institute Math. Sci., New York Univ., N.Y., 1974, 259 pp. · Zbl 0286.47037
[6]	Romanov V. G., “O lokalnoi razreshimosti nekotorykh mnogomernykh obratnykh zadach dlya uravnenii giperbolicheskogo tipa”, Dif. uravneniya, 25:2 (1989), 275-284
[7]	Romanov V. G., “Voprosy korrektnosti zadachi opredeleniya skorosti zvuka”, Sib. mat. zhurn., 30:4 (1989), 125-134 · Zbl 0712.76082
[8]	Romanov V. G., “O razreshimosti obratnykh zadach dlya giperbolicheskikh uravnenii v klasse funktsii, analiticheskikh po chasti peremennykh”, Dokl. AN SSSR, 304:4 (1989), 807-811 · Zbl 0682.35105
[9]	Durdiev D. K., “Mnogomernaya obratnaya zadacha dlya uravneniya s pamyatyu”, Sib. mat. zhurn., 35:3 (1994), 574-582 · Zbl 0859.35134
[10]	Durdiev D. K., “Some multidimensional inverse problems of memory determination in hyperbolic equations”, Zh. Mat. Fiz. Anal. Geom., 3:4 (2007), 411-423 · Zbl 1257.35191
[11]	Durdiev D. K., Safarov Zh. Sh., “Lokalnaya razreshimost zadachi opredeleniya prostranstvennoi chasti mnogomernogo yadra v integrodifferentsialnom uravnenii giperbolicheskogo tipa”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2012, no. 4(29), 37-47 · Zbl 1449.35461 · doi:10.14498/vsgtu1097