The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case
Sobolev type equations now constitute a vast area of nonclassical equations of mathematical physics. Those called nonclassical equations of mathematical physics, whose representation in the form of equations or systems of equations partial does not fit within one of the classical types (elliptic, parabolic or hyperbolic). In this paper we prove a generalized splitting theorem of spaces and actions of the operators for Sobolev type equations with respect to the relatively radial operator. The main research method is the Sviridyuk theory about relatively spectrum. Abstract results are applied to prove the unique solvability of the multipoint initial-final problem for the evolution equation of Sobolev type, as well as to explore the dichotomies of solutions for the linearized phase field equations.
Apart from the introduction and bibliography article comprises three parts. The first part provides the necessary information regarding the theory of p-radial operators, the second contains the proof of main result about generalized splitting theorem for strongly (L,p)-radial operator M. The third part contains the results of the application of the preceding paragraph for different tasks, namely to prove the unique solvability of the multipoint initial-final problem for Dzektser and to explore the dichotomies of solutions of the linearized phase field equations. References not purport to, and reflects only the authors' tastes and preferences.
1. Demidenko G.V., Uspenskii S.V. Partial differential equations and systems not solvable with respect to the highest-order derivative. New York Basel Hong Kong, Marcel Dekker, Inc., 2003. 239 p.
2. Gilmutdinova A.F. On Nonuniqueness of Solutions to the Showalter-Sidorov Problem for the Plotnikov Model (Russian). Vestnik of Samara State University. 2007, no. 9 /1, pp. 85-90.
3. Keller A.V. Relatively spectral theorem (Russian). Bulletin of the Chelyabinck State University, Series of Mathematic and Mechanic, 1996, no. 1 (3), pp. 62-66.
4. Manakova N.A., Dyl'kov A.G. Optimal Control of the Initial-Finish Problem for the Linear Hoff Model. Mathematical Notes, 2013, vol. 94, no. 1-2, pp. 220-230.
5. Sagadeeva M.A. Exponential Dichotomies of Solutions of a Class of Sobolev Type Equations (Russian) Vestnik Chelyabinsk State University, Seria 3, Mathematics. Mechanics. Informatics, 2003, no. 1, pp. 136-145.
6. Sagadeeva M.A. Dichotomy of the solutions for linear equations of Sobolev type. (Russian). Chelyabinsk, Publishing center of SUSU, 2012. 139 p.
7. Sagadeeva M.A., Shulepov A.N. The Approximations for Degenerate C0-semigroup (Russian) Bulletin of the South Ural State University. Series "Mathematical Modelling, Programming & Computer;Software". 2013, vol. 6, no. 2, pp. 133-137.
8. Sviridyuk G.A. On the General Theory of Operator Semigroups. (Russian) Uspekhi Mat. Nauk 49. 1994, vol. 49, no. 4(298), pp. 47-74 translation in Russian Math. Surveys. 1994, vol. 49, no. 4, pp. 45-74.
9. Sviridyuk G.A. Sobolev-type Linear Equations and Strongly Continuous Semigroups of Resolving Operators with Kernels. (Russian) Dokl. Akad. Nauk, 1994, vol. 337, no. 5, pp. 581-584 translation in Russian Acad. Set. Dokl. Math. 1995, vol. 50, no. 1, pp. 137-142.
10. Sviridyuk G.A., Keller A.V. Invariant Spaces and Dichotomies of Solutions of a Class of Linear Equations of Sobolev Type. (Russian) Izv. Vyssh. Uchebn. Zaved.Mat. 1997, no 5, pp. 60-68 translation in Russian Math. (Iz. VUZ), 1997, vol. 41, no. 5, pp. 57-65.
11. Sviridyuk G.A., Sukhanova M.V. Solvability of Cauchy Problem for Linear Singular Equations of Evolution Type. (Russian) Differential Equations, 1992, vol. 28, no. 3, pp. 508-515.
12. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht Boston Koln Tokyo, VSP, 2003.
13. Zagrebina S.A. The Multipoint Initial-Finish Problem for Hoff Linear Model. (Russian) Bulletin of the South Ural State University, Series "Mathematical Modelling, Programming & Computer;Software", 2012, no. 5 (264), pp. 4-12.
14. Zagrebina S.A. The Initial-Finite Problems for Nonclessical Models of Mathematical Physics. (Russian) Bulletin of the South Ural State University, Series "Mathematical Modelling, Programming & Computer;Software", 2013, vol. 6, no. 2, pp. 5-24.
15. Zagrebina S.A., Sagadeeva M.A. Generalized Showalter-Sidorov Problem for Sobolev Type Equations with strong (L,p)-radial operator. (Russian) Vestnik of Magnitogorsk State University, Mathematics, 2006, issue 9, pp. 17-27.
16. Zamyshlyaeva A.A., Yuzeeva A.V. The Initial-Finish Value Problem for the Boussinesque-Love Equation Defined on Graph (Russian) The Bulletin of Irkutsk State University, Series "Mathematics", 2010, vol. 3, no. 2, pp. 18-29.