Solvability of the first boundary value problem for a mixed type equation in a weight space

Abstract

The works of F. Tricomi, A. V. Bitsadze, M. M. Smirnov and many other authors are devoted to the study of various boundary value problems for equations of mixed type of second order. In these works, the theory of singular integral equations was used. Since the 1970s, functional methods and methods associated with functional analysis began to be applied to the study of boundary value problems for mixed type equations. The construction of a general theory of boundary value problems for equations of mixed type with an arbitrary variety of changing type began. In particular, under certain assumptions and the sign of the coefficient of the second derivative with respect to time near the bases of the cylindrical region, the existence and uniqueness of a regular solution to the enemy boundary value problem and the first boundary value problem for a second order mixed type equation is proved using the regularization method.

In 2019 A. N. Artyushin proved the existence and uniqueness of a generalized and regular solution to Vragov’s boundary value problem in the weighted Sobolev space, when the coefficient of the second derivative with respect to time can change sign on the bases of a cylindrical domain.

In this work, we will establish the existence of a generalized solution and the unique regular solvability of the first boundary value problem for a second order mixed type equation in the weighted Sobolev space, when the coefficient of the highest derivative of the equation with respect to time can change sign on the lower base and negative on the upper base of the cylindrical domain.