The Phase Space of the Modified Boussinesq Equation

We proved a unique solvability of the Cauchy problem for a class of semilinear Sobolev type equations of the second order. We used ideas and techniques developed by G.A. Sviridyuk for the investigation of the Cauchy problem for a class of semilinear Sobolev type equations of the first order and by A.A. Zamyshlyaeva for the investigation of the high-order linear Sobolev type equations. We also used theory of differential Banach manifolds which was finally formed in S. Leng's works. The initial-boundary value problem for the modified Bussinesq equation was considered as application. In article we considered two cases. The first one is when an operator L at the highest time derivative is continuously invertible. In this case for any point from a tangent fibration of an original Banach space there exists a unique solution lying in this space as trajectory. Particular attention was paid to the second case, when the operator L isn't continuously invertible and the Bussinesq equation is degenerate one. A local phase space in this case was constructed. The conditions for the phase space of the equation being a simple Banach manifolds are given.

Keywords

phase space, Sobolev type equation, relatively spectrally bounded operator, Banach manifold.

References

1. Wang S., Chen G. Small Amplitude Solutions of the Generalized IMBq Equation. Mathematical Analysis and Application, 2002, vol. 274, pp. 846 - 866.
2. Arkhipov D.G., Khabakhpashev G.A. The New Equation to Describe the Inelastic Interaction of Nonlinear Localized Waves in Dispersive Media. JTEP Letters, 2011, vol. 93, no. 8, pp. 423 - 426.
3. Sviridyuk G.А., Sukacheva, T.G. The Phase Space of a Class of Operator Equations of Sobolev Type [Fazovye prostranstva odnogo klassa operatornykh uravneniy tipa Soboleva]. Differential Equation, 1990, vol. 26, no. 2, pp. 250 - 258.
4. Zagrebina S.А. On the Showalter-Sidorov Problem. Russian Mathematics, 2007, vol. 51, no. 3, pp. 19 - 24.
5. Keller А.V. Numerical Solution of the Starting Control System of Equations of Leontiev Type [Chislennoe reshenie zadachi startovogo upravleniya dlya sistemy uravneniy leont'evskogo tipa]. Obozrenie prikladnoy i promyshlennoy matematiki [Review of Industrial and Applied Mathematics], 2009, vol. 16, no. 2. pp. 345 - 346.
6. Manakova N.А. Optimal Control Problem for the Oskolkov Nonlinear Filtration Equation. Differential equations, 2007, vol. 43, no. 9, pp. 1213 - 1221.
7. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, K'oln, Tokyo, VSP, 2003.
8. Leng S. Introduction to Differentiable Manifolds. New-York, Spriger, Verlag, 2002.
9. Nirenberg L. Topics in Nonlinear Functional Analysis. New ed., New-York, 2001.
10. Hassard B.D. Theory and Applications of Hopf Bifurcation. Cambridge University Press, 1981.
11. Sviridyuk G.А., Kazak V.О. The Phase Space of one Generalized Model by Oskolkov. Siberian Mathematical journal, 2003, vol. 44, no 5, pp. 877 - 882.