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 highorder 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.