The Cauchy problem for the Sobolev type equation of higher order

Of concern is the semilinear mathematical model of ion-acoustic waves in plasma. It is studied via the solvability of the Cauchy problem for an abstract complete semilinear Sobolev type equation of higher order. The theory of relatively polynomially bounded operator pencils, the theory of differentiable Banach manifolds, and the phase space method are used. Projectors splitting spaces into direct sums and an equation into a system of two equivalent equations are constructed. One of the equations determines the phase space of the initial equation, and its solution is a function with values from the eigenspace of the operator at the highest time derivative. The solution of the second equation is the function with values from the image of the projector. Thus, the sufficient conditions were obtained for the solvability of the problem under study. As an application, we consider the fourth-order equation with a singular operator at the highest time derivative, which is in the base of mathematical model of ion-acoustic waves in plasma. Reducing the model problem to an abstract one, we obtain sufficient conditions for the existence of a unique solution.