Linear inverse problems for a class of equations of Sobolev type

A. I. Kozhanov ; G. V. Namsaraeva

Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 2, pp. 153-171.

Résumé

We study the solvability of inverse problems of finding together with the solution $u(x,t)$ also an unknown factor $q(t)$ in equation $$D^{2p}_t(u-\Delta u)+Bu=f_0(x,t)+q(t)h_0(x,t)$$ ($t\in (0,T)$, $x\in\Omega\subset \mathbb{R}^n$, $p$ is a natural number, $D^k_t=\frac{\partial^k}{\partial t^k}$, $\Delta$ is the Laplace operator with respect to the spatial variables, $B$ is a linear second-order differential operator, acting also on the spatial variables, $f_0(x,t)$ and $h_0(x,t)$ are given functions). Integral overdetermination condition is used as an additional condition in these problems. The existence and uniqueness theorems for regular solutions (i. e. having all the generalized derivatives in the sense of S.L. Sobolev, presenting in the equation) are proved.

Détail

Keywords: Sobolev type equation, inverse problem, unknown right-hand side, integral overdetermination, regular solution, solution existence, solution uniqueness.

Bibliographie

[1] G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974, 635 pp. | MR | Zbl

[2] H. Ikezi, “Experimental research of solitons in plasma”, Solitons in Action, Academic Press, New York, 1978, 153–172

[3] Demidenko G.V., Uspenskii S.V., Partial differential equations and systems not solvable with respect to the highest-order derivative, Marcel Dekker, Inc., New York, Basel, 2003, 481 pp. | MR | MR | Zbl

[4] Megraliev Ya.T., “The inverse boundary problem for the Boussinesq — Love equation with the additional integral condition”, Siberian journal of industrial mathematics, 16:1 (2013), 75–83 (In Russ.) | MR | Zbl

[5] Megraliev Ya.T., Alizade F.Kh., “An inverse boundary value problem for a fourth-order Boussinesq equation with an integral condition”, Chebyshevskii Sbornik, 14:4 (2013), 167–179 (In Russ.) | MR

[6] Kasymalieva A.A., Inverse problems for Boussinesq — Love equation, PhD Thesis, Bishkek, 2014 (In Russ.)

[7] Namsaraeva G.V., “Linear inverse problems for some analogs of the Boussinesq equation”, Mathematical Notes of North-Eastern Federal University, 21:2 (2014), 47–59 (In Russ.) | Zbl

[8] Namsaraeva, G.V., “Inverse problems of recovering external sources in the equation of longitudinal wave propagation”, Journal of Applied and Industrial Mathematics, 10:3 (2016), 386–396 | DOI | MR | Zbl

[9] Trenogin V.A., Functional Analysis, Nauka Publ., Moscow, 1980, 496 pp. (In Russ.) | MR

[10] Yakubov S.Ya., Linear differential-operator equations and their application, Elm Publ., Baku, 1985, 220 pp. (In Russ.)

[11] Amirov Sh., Kozhanov A.I., “Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation”, Mathematical Notes, 99:1–2 (2016), 183–191 | DOI | DOI | MR | Zbl

[12] Utkina E.A., “Dirichlet problem for a fourth-order equation”, Differential Equations, 47:4 (2011), 599–603 | DOI | MR | Zbl

[13] Utkina E.A., “Uniqueness of the solution of the Dirichlet problem for an $n$-dimensional pseudoparabolic equation”, Differential Equations, 48:10 (2012), 1423–1428 | DOI | MR | Zbl

[14] Kozhanov A.I., Pinigina N.R., “Boundary-value problems for some higher-order nonclassical differential equations”, Mathematical Notes, 101:3–4 (2017), 467–474 | DOI | DOI | MR | Zbl

[15] Ladyzhenskaya O.A., Ural’tseva N.N., Linear and Quasilinear Equations of Elliptic Type, Nauka Publ., Moscow, 1973, 578 pp. (In Russ.) | MR