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.