Volume 7, no. 4Pages 126 - 131

The Lyapunov Stability of the Cauchy-Dirichlet Problem for the Generalized Hoff Equation

P.O. Moskvicheva, I.N. Semenova
We consider the initial boundary value problem with homogeneous Dirichlet boundary conditions for the generalized Hoff equation in a bounded domain. This equation models the dynamics of buckling of a double-tee girder under constant load and belongs to a large class of Sobolev type semilinear equations (We can isolate the linear and non-linear parts of the operator acting on the original function). The paper addresses the stability of zero solution of this problem. There are two methods in the theory of stability: the first one is the study of stability by linear approximation and the second one is the study of stability by Lyapunov function. We use the second Lyapunov's method adapted to the case of incomplete normed spaces. The main result of this paper is a theorem on the stability and asymptotic stability of zero solution to this problem.
Full text
Keywords
Sobolev-type equation; phase space; Lyapunov stability.
References
1. Hoff N.J. Creep Buckling. Aeronautical Quarterly, 1956, vol. 7, no. 1, pp. 1-20.
2. Sidorov N.A. Obshhie voprosy reguljarizacii v zadachah teorii vetvlenija [Common Questions of Regularity in Problems of Ramification Theory]. Irkutsk, Irkutsk Gos. Univ. Publ., 1982. 314 p. (in Russian)
3. Sidorov N.A., Romanova O.A. [Application of Certain Results of Branching Theory in the Solution of Degenerate Differential Equations.] Differetial'niye Uravneniya [Differential Equations], 1983, vol. 19, no. 9, pp. 1516-1526. (in Russian)
4. Sidorov N.A., Falaleev M.V. [Generalized Solutions of Differential Equations with a Fredholm Operator at the Derivative]. Differetial'niye Uravneniya [Differential Equations], 1987, vol. 23, no. 4, pp. 726-728. (in Russian)
5. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. tRussian Academy of Sciences. Izvestiya Mathematics, 1994, vol. 42, no. 3, pp. 601-614. DOI: 10.1070/IM1994v042n03ABEH001547
6. Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 2, pp. 262-266. DOI: 10.1023/A:1013919500605
7. Sviridyuk G.A., Trineeva I.K. A Whitney Fold in the Phase Space of the Hoff Equation. Russian Mathematics, 2005, vol. 49, no. 10, pp. 49-55.
8. Bajazitova A.A. [The Phase Space of the Initial-Boundary Value Problem for a Generalized Hoff Equation]. Vestnik MaGU. Matematika [Bulletin of the Magnitogorsk State University], 2010, vol. 12, pp. 15-21. (in Russian)
9. Zagrebina S.A., Pivovarova P.O. [Stability and Instability of the Solutions of the Hoff Equations. The Numerical experiment]. Nonclassical Equations of Mathematical Physics, Novosibirsk, Izdatel'stvo Instituta Matematiki Im. S.L. Soboleva SO RAN, 2010, pp. 88-94. (in Russian)
10. Sviridyuk G.A., Semenova I.N. Solvability of an Inhomogeneous Problem for a Generalized Boussinesq Filtration Equation. Differential Equations, 1988, vol. 24, no. 9, рр. 1065-1069.