### Abstract:

Boundary-value problems for parabolic equations іп domains with moving boundaries are fundamentally different from the classical parabolic equations. Due to the dependence of the region size on time, the methods of separation of variables and integral transformations are not applicable to this type of problems іn general case, since remaining within the framework of classical methods of mathematical physics, it is
not possible to coordinate the solution of the heat conduction equation with the motion of the boundary of the heat transfer region. The solution of this problem has been the subject of research of many domestic and foreign mathematicians [1-8]. A large number of works are devoted to boundary-value problems in non-degenerate domains; they considered the existence of classical solutions by the method of thermal potentials for both the heat conduction equation and for more general parabolic equations. But if the region degenerates at the initial moment of time, then the method of successive approximations for solving integral equations cannot be applied. Since at the degeneration of the domain integral operators become special, that is, when they affect the constant and the upper limit tends to zero, they do not tend to zero. Integral equations of this kind were obtained in [8] in the study of the thermal field of liquid contact bridges and an asymptotic solution was found that can be used to solve practical problems. This paper is devoted to the study of the first boundary value problem for the heat conduction equation with a discontinuous coefficient in the domain that degenerates at the initial moment of time when the boundary moves by linear law. An explicit form of the solution of this problem is obtained, afterwards that can be applied for a numerical approximations.