Nonlocal problem for a class of third-order equations

This article considers a nonlocal problem in a cylindrical domain for the third-order mixed-composite type equation of the form
$$u_{ttt} - µ(x_1)\frac{\partial}{\partial(x_1)} \Delta u - a(x, t) \Delta u = f(x, t),$$
where $x_1µ(x_1) > 0$ for $x_1 \ne 0, µ(0) = 0, x = (x_1, x_2, . . . , x_n) \in R_n$. Using the Galerkin method, it is proved that this nonlocal problem, under certain conditions on the coefficients and the right side of the equation, has a unique solution in Sobolev spaces. The proof is based on the Galerkin method with the choice of a special basis and a priori estimates. New theorems are also proved regarding the existence and uniqueness of the solution of the nonlocal problem, which allow expanding the range of solvable problems in the theory of boundary value problems for nonclassical equations of mathematical physics.