Boundary value problems with conjugation conditions for quasi-parabolic equations of the third order with a discontinuous sign–variable coefficient

The aim of this work is to study the solvability in Sobolev spaces of boundary value problems for third order differential equations with a discontinuous sign–variable coefficient at the highest derivative with respect to the time variable. Since the equation has a discontinuous leading coefficient, in addition to setting the boundary conditions it is also necessary to set some conjugation conditions. For the problems under study, existence and uniqueness theorems are proved for the class of regular solutions, i.e., for the solutions that have all Sobolev weak derivatives up to the third order in time variable and up to the second order in spatial variables.