SOLVABILITY OF THE INVERSE PROBLEM FOR THE PSEUDOHYPERBOLIC EQUATION
DOI:
https://doi.org/10.26577/JMMCS.2022.v115.i3.01Keywords:
Pseudohyperbolic equation, nverse problem, Klein-Gordon equation, Galerkin method, compactness method, existence, uniquenessAbstract
This paper investigates the solvability of the inverse problem of finding a solution and an unknown coefficient in a pseudohyperbolic equation known as the Klein-Gordon equation. A distinctive feature of the given problem is that the unknown coefficient is a function that depends only on the time variable. The problem is considered in the cylinder, the conditions of the usual initial-boundary value problem are set. The integral overdetermination condition is used as an additional condition. In this paper, the inverse problem is reduced to an equivalent problem for the loaded nonlinear pseudohyperbolic equation. Such equations belong to the class of partial differential equations, not resolved with respect to the highest time derivative, and they are also called composite type equations. The proof uses the Galerkin method and the compactness method (using the obtained a priori estimates). For the problem under study, the authors prove existence and uniqueness theorems for the solution in appropriate classes.
References
[2] Dodd R. K. Solitons and nonlinear wave equations / R. K. Dodd, I. C. Eilbeck, J. D. Gibbon,H. C. Morris. L.: Acad.
Press, 1982.
[3] Strauss W, Vazquez L. Numerical solution of a nonlinear Klein-Gordon equation.JComput Phys. 1978;28(2):271-278.
[4] Jim nez S, Vazquez L. Analysis of four numerical schemes for a nonlinear Klein-Gordon equation.Appl Math Comput. 1990;35(1):61-94.
[5] Wong YS, Qianshun C, Lianger G. An initial-boundary value problem of a nonlinear Klein-Gordon equation.Appl Math Comput.1997;84(1):77-93.
[6] Wazwaz A. M. Compactons, solitons and periodic solutions for some forms of nonlinear Klein-Gordon equations // Chaos Solitons Fractals. 2006. V. 28. P. 1005–1013.
[7] Sassaman R., Biswas A. Soliton perturbation theory four phi-four model and nonlinear Klein-Gordon equations // Comm. Nonlinear Sci. Numer. Simulat. 2009. V. 14. P. 3239–3249.
[8] Sassaman R., Biswas A. Topological and non-topological solitons of the generalized Klein-Gordon equations // Appl. Math. Comput. 2009. V. 215. P. 212–220.
[9] Bratsos A. G., Petrakis L. A. A modified predictor-corrector scheme for the Klein-Gordonequation // Intern. J. Comput. Math. 2010. V. 87, N 8. P. 1892–1904.
[10] B¨ohme C, Michael R. A scale-invariant Klein-Gordon model with time-dependent potential.Ann Univ Ferrara.
2012;58(2):229-250. [11] Bhme C, Michael R. Energy bounds for Klein-Gordon equations with time-dependent potential.Ann Univ Ferrara. 2013;59(1):31-55.
[12] Hao Cheng, Xiyu Mu, GreenTMs function for the boundary value problem of the static Klein-Gordon equation stated on a rectangular region and its convergence analysis. Boundary Value Problems (2017) 2017:72.
[13] Tekin I, Mehraliyev YT, Ismailov MI. Existence and uniqueness of an inverse problemfor nonlinear Klein-Gordon equation.Math Meth Appl Sci. 2019;1-15.https://doi.org/10.1002/mma.5609
[14] Sobolev, S. L.; On certain new problem of mathematical physics. Izvestia Acad. Nauk.Mathem. 1954, no. 18, p. 3-50.
[15] Kozhanov, A.I., Comparison Theorems and Solvability of Boundary Value Problems for Some Classes of Evolution Equations Like Pseudoparabolic and Pseudohyperbolic Ones, Preprint Inst. Mat. Acad. Sci., Novosibirsk, 1990, no. 17
[16] Demidenko, G., The Cauchy problem for pseudohyperbolic equations, Selcuk J Appl Math, Vol. 1,No.1, pp.47-62, 2000.
[17] Sveshnikov, A. G.; Alshin, A. B.; Korpusov, O. M.; Pletner, Yu. D.; Linear and nonlinearSobolev type equations.
Moscow.,Phizmatlit. 2007.
[18] Korpusov, O. M., Blow-up in nonclassical wave equations. Moscow, URSS. 2010.
[19] Pulkina, L.S. Solution to nonlocal problems of pseudohyperbolic equations // Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 116, pp. 1–9.
[20] Yamamoto, M., Zhang, X.: Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns. Appl. Math. Optim. 48, 211–228 (2003)
[21] Oussaeif,T.E., Bouziani, A.: Inverse problem of hyperbolic equationwith an integral overdetermination condition. Electron J. Diff. Equ. 2016(138), 1–7 (2016)
[22] Shahrouzi, M.: On behavior of solutions to a class of nonlinear hyperbolic inverse source problem. Acta. Math. Sin-English Ser. 32(6), 683–698 (2016)
[23] Shahrouzi, M.: Blow up of solutions to a class of damped viscoelastic inverse source problem. Differ. Equ. Dyn. Syst. 28, 889-899 (2020)
[24] Shahrouzi, M.: General decay and blow up of solutions for a class of inverse problem with elasticity term and variableexponent nonlinearities. Math Meth Appl Sci, (2021), Early view, https://doi.org/10.1002/mma.789
