Boundary problems for a class of differential equations with multiple characteristics

In this paper we study differential equations with multiple characteristics (differential equations of the composite type) of the form
$$\frac{\partial^3}{\partial x^3}\left(u_t-\alpha u_x \right)+\beta \Delta_y u + \gamma u=f(x,y,t),$$
($\alpha$, $\beta$, $\gamma$ are constants). For these equations, we propose the formulation of new boundary-value problems, for the proposed problems we prove the existence and uniqueness theorems for regular solutions (having all the generalized in the sense of S.L. Sobolev derivatives entering the equation). The technique of proof is based on the regularization method. The equations studied in this paper represent, by their structure, equations that are called in the literature by equations that are not resolved with respect to the derivative. Some possible generalizations are indicated for the problems under study.