Abstract
We consider a class of strictly pseudohyperbolic equations and establish some solvability conditions of the Cauchy problem in the class of weighted Sobolev spaces. We also prove the uniqueness of solutions and obtain some estimates.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Demidenko G.V. and Uspenskii S.V., Partial Differential Equations and Systems Not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York and Basel (2003).
Sobolev S.L., Selected Works. Vols. I and II, Inst. Mat. Sibirsk. Otdel. Akad. Nauk and Filial “Geo” Sibirsk. Otdel. RAN, Novosibirsk (2003) [Russian].
Favini A. and Yagi A., Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, Basel, and Hong Kong (1999).
Sveshnikov A.G., Alshin A.B., Korpusov M.O., and Pletner Yu.D., Linear and Nonlinear Equations of Sobolev Type, Fizmatlit, Moscow (2007) [Russian].
Sviridyuk G.A. and Fedorov V.E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, and Köln (2003).
Korpusov M.O., Blow-up in Nonclassical Nonlinear Equations, Librokom, Moscow (2011).
Demidenko G., “The Cauchy problem for pseudohyperbolic equations,” Selčuk J. Appl. Math., vol. 1, no. 1, 47–62 (2001).
Demidenko G.V., “Solvability conditions of the Cauchy problem for pseudohyperbolic equations,” Sib. Math. J., vol. 56, no. 6, 1028–1041 (2015).
Leray J., Hyperbolic Differential Equations, Institute for Advanced Study, Princeton (1953).
Petrowsky I.G., Selected Works. Part 1: Systems of Partial Differential Equations and Algebraic Geometry, Gordon and Breach, Amsterdam (1996).
Galpern S.A., “The Cauchy problem for general systems of linear partial differential equations,” Uspekhi Mat. Nauk, vol. 18, no. 2, 239–249 (1963).
Fedotov I. and Volevich L.R., “The Cauchy problem for hyperbolic equations not resolved with respect to the highest time derivative,” Russian J. Math. Phys., vol. 13, no. 3, 278–292 (2006).
Demidenko G.V., “The Cauchy problem for generalized S.L. Sobolev equations,” in: Functional Analysis and Mathematical Physics. Russian, Inst. Mat., Novosibirsk (1985), 88–105.
Demidenko G.V., “On quasielliptic operators in \( {}_{n} \),” Sib. Math. J., vol. 39, no. 5, 884–893 (1998).
Uspenskii S.V., “The representation of functions defined by a certain class of hypoelliptic operators,” Proc. Steklov Inst. Math., vol. 117, 343–352 (1972).
Hardy G.H., Littlewood J.E., and Pólya G., Inequalities, Cambridge University, Cambridge (1988).
Funding
The research was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2023, Vol. 64, No. 5, pp. 895–911. https://doi.org/10.33048/smzh.2023.64.502
To the blessed memory of Sergei L’vovich Sobolev.
Rights and permissions
About this article
Cite this article
Bondar, L.N., Demidenko, G.V. On Well-Posedness of the Cauchy Problem for Pseudohyperbolic Equations in Weighted Sobolev Spaces. Sib Math J 64, 1076–1090 (2023). https://doi.org/10.1134/S0037446623050026
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446623050026
Keywords
- equations that are not solved with respect to the higher-order derivative
- pseudohyperbolic equations
- energy estimate