Volume 16, no. 3Pages 65 - 73
Stability of a Stationary Solution to One Class of Non-Autonomous Sobolev Type EquationsA.V. Buevich, M.A. Sagadeeva, S.A. Zagrebina
The article is devoted to the study of the stability of a stationary solution to the Cauchy problem for a non-autonomous linear Sobolev type equation in a relatively bounded case. Namely, we consider the case when the relative spectrum of the equation operator can intersect with the imaginary axis. In this case, there exist no exponential dichotomies and the second Lyapunov method is used to study stability. The stability of stationary solutions makes it possible to evaluate the qualitative behavior of systems described using such equations. In addition to introduction, conclusion and list of references, the article contains two sections. Section 1 describes the construction of solutions to non-autonomous equations of the class under consideration, and Section 2 examines the stability of a stationary solution to such equations.Full text
- relatively bounded operator; Lyapunov’s second method; local stream of operators; asymptotic stability.
- 1. Sviridyuk G.A., Manakova N.A. Nonclassical Mathematical Physics Models. Phase Space of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31–51. DOI: 10.14529/mmph160304 (in Russian)
2. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, de Gruyter, 2011.
3. Sell G.R. Topological Dynamics and Ordinary Differential Equations. London, Van Nostrand Reinhold, 1971.
4. 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.
5. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, VSP, 2003.
6. Keller A.V. The Leontief’s Type Systems: Classes of Problems with the Showalter – Sidorov Intial Condition and Numerical Solving. The Bulletin of Irkutsk State University. Series Mathematics, 2010, vol. 3, no. 2, pp. 30–43. (in Russian)
7. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Russian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601–614. DOI: 10.1070/IM1994v042n03ABEH001547
8. Sagadeeva M.A. Investigation of Solutions Stability for Linear Sobolev Type Equations. PhD (Math) Thesis. Chelyabinsk, 2006. (in Russian)
9. Sagadeeva M.A. Degenerate Flows of Solving Operators for Nonstationary Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 1, pp. 22–30. DOI: 10.14529/mmph170103 (in Russian)
10. Zagrebina S.A., Moskvicheva P.O. Stability in Hoff Models. Saarbrucken, LAMBERT Academic Publishing, 2012. (in Russian)
11. Moskvicheva P.O. Stability of the Evolutionary Linear Sobolev Type Equation. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2017, vol. 9, no. 3, pp. 13–17. (in Russian) DOI: 10.14529/mmph170302