The Cauchy Problem for the Sobolev Type Equation of Higher Order

A. A. Zamyshlyaeva; E. V. Bychkov

doi:10.14529/mmp180101

Authors

A. A. Zamyshlyaeva South Ural State University
E. V. Bychkov South Ural State University

DOI:

https://doi.org/10.14529/mmp180101

Keywords:

semilinear Sobolev type equation of higher order, Cauchy condition, relatively polynomially bounded operator pencils, phase space method.

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 dierentiable 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 sucient conditions were obtained for the solvability of the problem under study. As an application, we consider the fourthorder 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 sucient conditions for the existence of a unique solution.

Author Biographies

A. A. Zamyshlyaeva, South Ural State University

Doctor of Physico-Mathematical Sciences, Docent

E. V. Bychkov, South Ural State University

Candidate of Physico-Mathematical Sciences

References

Zamyshlyaeva A.A., Muravyev A.S. Computational Experiment for One Mathematical Model of Ion-Acoustic Waves. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 127-132.

Sviridyuk G.A., Sukacheva T.G. [The Phase Space of a Class of Operator Equations of Sobolev Type]. Differential Equations, 1990, vol. 26, no. 2, pp. 250-258. (in Russian)

Zamyshliaeva A.A. Lineynye uravneniya sobolevskogo tipa vysokogo poryadka [Linear Sobolev Type Equations of Higher Order]. Chelyabinsk, Publishing center of SUSU, 2012. (in Russian)

Sviridyuk G.A., Sukacheva T.G. Galerkin Approximations of Singular Nonlinear Equations of Sobolev Type. Russian Mathematics, 1989, vol. 33, no. 10. pp. 56-59.

Manakova N.A., Bogatyreva E.A. On a Solution of the Dirichlet-Cauchy Problem for the Barenblatt-Gilman Equation. The Bulletin of Irkutsk State University. Series: Mathematics, 2014, vol. 7, pp. 52-60. (in Russian)

Manakova N.A., Dyl'kov A.G. Optimal Control of the Solutions of the Initial-Finish Problem for the Linear Hoff Model. Mathematical Notes, 2013, vol. 94, no. 1-2, pp. 220-230. DOI: 10.1134/S0001434613070225

Keller A.V., Sagadeeva M.A. The Optimal Measurement Problem for the Measurement Transducer Model with a Deterministic Multiplicative Effect and Inertia. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2014, vol. 7, no. 1, pp. 134-138. DOI: 10.14529/mmp140111 (in Russian)

Sviriduyk G.A., Zamyshlyaeva A.A. The Phase Space of a Class of Linear Higher Order Sobolev Type Equations. Dierential Equations, 2006, vol. 42, no 2, pp. 269-278. DOI: 10.1134/S0012266106020145

Leng S. Introduction to Differentiable Manifolds. N.Y., Springer, 2002.

Zamyshlyaeva A.A. The Phase Space of a High Order Sobolev Type Equation. The Bulletin of Irkutsk State University. Series: Mathematics, 2011, vol. 4, no. 4, pp. 45-57. (in Russian)

Nirenberg L. Topics in Nonlinear Functional Analysis. N.Y., American Mathematical Society, 2001.