Regularization of the boundary value problem for an inhomogeneous high-order parabolic equation with one degeneracy line

Abstract

In this paper, we study an ill-posed initial-boundary value problem for a high-order inhomogeneous parabolic equation with one degeneracy line. We obtain a priori estimate of the solution of the problem, determine the correctness set, and prove the uniqueness and conditional stability of the solution in this set. As a result, we construct a regularized approximate solution in the correctness set according to the theory of A.N. Tikhonov. In this case, an approximate solution is constructed depending on the evenness or oddness of the parameter $n$, which determines the order of the equation.