Classical solution of the mixed problem for the string oscillation equation with linear differential polynomials in boundary conditions

The proof of the well-posedness of the mixed problem for the string oscillation equation in the half-strip with differential polynoms in the boundary conditions. The conditions of the existence of the unique and smooth enough solution are obtained in the half strip in general. It is shown that it is reduced to the solution of the initial-value problems for the ordinary linear differential equations with variable coefficients. The case when the solution smoothness is reduced during the increasing of the time and the case when it doesn’t happen are studied. For both cases the sufficient conditions for smooth reduction (conservation) are obtained. These conditions are based on the coefficients in boundary conditions. Also, with the help of the characteristics method the necessary and sufficient matching conditions are obtained. These conditions guarantee the existence and uniqueness of the classical solution of the given problem when given functions are smooth enough. The obtained results are given for both homogeneous initial equation and inhomogeneous one.