On the Solvability of One Class of Third-Order Differential Equations

We consider a one-dimensional mixed problem for one class of third-order partial differential equations with nonlinear right-hand side. The concept of generalized solution of this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solutions for this problem is reduced to the problem of solvability of a countable system of nonlinear integrodifferential equations. By using Bellman's inequality, we prove the uniqueness of generalized solution. Under certain conditions imposed on the initial functions and the right-hand side of the equation, the existence theorem is proved for the generalized solution by the method of successive approximations.