The Cauchy Problem for the Second Order Semilinear Sobolev Type Equation

We prove a unique solvability of the Cauchy problem for a class of second order semilinear Sobolev type equations. We use ideas and techniques developed by G.A. Sviridyuk for the investigation of the Cauchy problem for a class of first order semilinear Sobolev type equations and by A.A. Zamyshlyaeva for the investigation of the high-order linear Sobolev type equations. We also use the theory of differential manifolds following, say, Lang’s books. In the article we consider two cases.
The first one is where an operator A at the highest time derivative is continuously invertible. In this case for any point from the tangent bundle of the original Banach space there exists a unique solution lying in this space as a trajectory. The second case, where the operator A is not continuously invertible, is of great interest for us. Here we use the phase space method. It consists in reducing a singular equation to a regular one which is defined on a subset of the original Banach space consisting of admissible initial values which is understood as a phase space. Under the condition of polynomial boundedness of operator pencil in the case where infinity is a removable singularity of its A-resolvent, a set, which
is locally a phase space of the original equation, is constructed. In the last section the abstract theory is applied to an initial-boundary value problem for Boussinesque – L¨ove equation.