Abstract:
Of concern is the semilinear mathematical model of ion-acoustic waves in plasma. It is studied via the solvability of the Cauchy problem for an abstract complete semilinear Sobolev type equation of higher order. The theory of relatively polynomially bounded operator pencils, the theory of differentiable Banach manifolds, and the phase space method are used. Projectors splitting spaces into direct sums and an equation into a system of two equivalent equations are constructed. One of the equations determines the phase space of the initial equation, and its solution is a function with values from the eigenspace of the operator at the highest time derivative. The solution of the second equation is the function with values from the image of the projector. Thus, the sufficient conditions were obtained for the solvability of the problem under study. As an application, we consider the fourth-order equation with a singular operator at the highest time derivative, which is in the base of mathematical model of ion-acoustic waves in plasma.
Reducing the model problem to an abstract one, we obtain sufficient conditions for the existence of a unique solution.
Keywords:
semilinear Sobolev type equation of higher order; Cauchy condition; relatively polynomially bounded operator pencils; phase space method.
Citation:
A. A. Zamyshlyaeva, E. V. Bychkov, “The Cauchy problem for the Sobolev type equation of higher order”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 11:1 (2018), 5–14
\Bibitem{ZamByc18}
\by A.~A.~Zamyshlyaeva, E.~V.~Bychkov
\paper The Cauchy problem for the Sobolev type equation of higher order
\jour Vestnik YuUrGU. Ser. Mat. Model. Progr.
\yr 2018
\vol 11
\issue 1
\pages 5--14
\mathnet{http://mi.mathnet.ru/vyuru413}
\crossref{https://doi.org/10.14529/mmp180101}
\elib{https://elibrary.ru/item.asp?id=32711843}
Linking options:
https://www.mathnet.ru/eng/vyuru413
https://www.mathnet.ru/eng/vyuru/v11/i1/p5
This publication is cited in the following 12 articles:
A. V. Keller, “O napravleniyakh issledovanii uravnenii sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 16:4 (2023), 5–32
N. A. Manakova, O. V. Gavrilova, K. V. Perevozhikova, “Semilinear models of sobolev type. Non-uniqueness of solution to the Showalter–Sidorov problem”, Vestnik YuUrGU. Ser. Mat. Model. Progr., 15:1 (2022), 84–100
A. A. Zamyshlyaeva, E. V. Bychkov, “Polulineinye matematicheskie modeli sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:1 (2022), 43–59
O. G. Kitaeva, “Invariantnye mnogoobraziya polulineinykh uravnenii sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:1 (2022), 101–111
D. E. Shafranov, “Uravneniya sobolevskogo tipa v prostranstvakh differentsialnykh form na rimanovykh mnogoobraziyakh bez kraya”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 15:1 (2022), 112–122
E. V. Bychkov, “Analytical study of the mathematical model of wave propagation in shallow water by the Galerkin method”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 14:1 (2021), 26–38
K. Yu. Kotlovanov, “On one equation of internal waves”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 13:2 (2021), 11–16
Alyona Zamyshlyaeva, Aleksandr Lut, “Inverse Problem for the Sobolev Type Equation of Higher Order”, Mathematics, 9:14 (2021), 1647
Evgeniy Bychkov, Georgy Sviridyuk, Alexey Bogomolov, “Optimal control for solutions to Sobolev stochastic equations”, ejde, 2021:01-104 (2021), 51
A. O. Kondyukov, T. G. Sukacheva, “Fazovoe prostranstvo pervoi nachalno-kraevoi zadachi dlya sistemy Oskolkova vysshego poryadka”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 11:4 (2018), 67–77
A. A. Zamyshlyaeva, O. N. Tsyplenkova, “Optimal control of solutions to the Showalter–Sidorov problem in a model of linear waves in plasma”, J. Comp. Eng. Math., 5:4 (2018), 46–57
E. V. Bychkov, “Finite difference method for modified Boussinesq equation”, J. Comp. Eng. Math., 5:4 (2018), 58–63
Citation in format AMSBIB
Contact us: