On the boundary value problem for the loaded parabolic equations with irregular coefficients

Abstract

In the paper we consider the generalized solvability of boundary value problem for the loaded parabolic
equations with irregular coefficients. Theorem on unique solvability of the boundary value problem is
proved. The correctness of the theorem and the accuracy of selected functional spaces are established by
obtained a priori estimates. The proof of the theorem is carried out using the theory of Sobolev spaces, the
method of a priori estimates, and the Galerkin method. Along with the initial boundary value problem,
the corresponding adjoint boundary value problem is investigated. To prove the solvability of the adjoint
problem, we define a linear continuous form and use the duality relations.