A class of second order Sobolev type equations and degenerate groups of operators

Initial boundary value problems for inhomogeneous partial differential equations unsolved with respect to second time derivative with polynomials of elliptic selfadjoint high order differential operator with respect to spatial variables is studied. Using reduction to system of first order equations and methods of degenerate semigroups of operators theory solvability conditions of considered problem is found. Obtained results are illustrated on the examples of Boussinesq equation and Boussinesq–Love equation.