Abstract
The Cauchy problem for one pseudohyperbolic system is considered. The unique solvability of this problem in Sobolev spaces is proved. Systems of this type arise in describing wave dynamics in rods.
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REFERENCES
G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative (Nauchnaya Kniga, Novosibirsk, 1998; Marcel Dekker, New York, Basel, 2003).
S. I. Gerasimov and V. I. Erofeev, Problems of Wave Dynamics for Structural Elements (Ross. Fed. Yad. Tsentr-Vseross. Nauchn.-Issled. Inst. Eksp. Fiz., Sarov. 2014) [in Russian].
I. S. Rao, Advanced Theory of Vibration (Wiley, New York, 1992).
G. V. Demidenko, “Solvability conditions of the Cauchy problem for pseudohyperbolic equations,” Sib. Math. J. 56 (6), 1028–1041 (2015).
G. V. Demidenko, “On solvability of the Cauchy problem for pseudohyperbolic equations,” Sib. Adv. Math. 11 (4), 25–40 (2001).
I. Fedotov and L. V. Volevich, “The Cauchy problem for hyperbolic equations not resolved with respect to the highest time derivative,” Russ. J. Math. Phys. 13 (3), 278–292 (2006).
I. Fedotov, M. Shatalov, and J. Marais, “Hyperbolic and pseudo-hyperbolic equations in the theory of vibration,” Acta Mech. 227 (11), 3315–3324 (2016).
O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations of Functions and Imbedding Theorems (Nauka, Moscow, 1975; Wiley, New York, 1978).
S. K. Godunov, Ordinary Differential Equations with Constant Coefficients, Vol. 1: Boundary Value Problems (Novosibirsk. Gos. Univ., Novosibirsk, 1994) [in Russian].
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1976; Dover, New York, 1999).
G. V. Demidenko, “On weighted Sobolev spaces and integral operators defined by quasi-elliptic equations,” Dokl. Akad. Nauk 334 (4), 420–423 (1994).
G. V. Demidenko, “On quasielliptic operators in \({{R}_{n}}\),” Sib. Math. J. 39 (5), 884–893 (1998).
G. V. Demidenko, “Isomorphic properties of one class of differential operators and their applications,” Sib. Math. J. 42 (5), 865–883 (2001).
Funding
This work was supported by the Mathematical Center in Akademgorodok under agreement no. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation.
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Dedicated to Academician S.K. Godunov on the occasion of his 90th birthday
Translated by N. Berestova
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Bondar, L.N., Demidenko, G.V. & Pintus, G.M. Cauchy Problem for One Pseudohyperbolic System. Comput. Math. and Math. Phys. 60, 615–627 (2020). https://doi.org/10.1134/S0965542520040053
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DOI: https://doi.org/10.1134/S0965542520040053