Perturbed initial boundary value conjugation problems

The non-self-adjoint boundary value, spectral and initial boundary value problems generated by the sesquilinear form were considered in the author's previous works. Such problems were studied in the case of one domain. The existence of weak solutions of the boundary value problems were obtained. Based on these boundary value problems, non-self-adjoint spectral problems were investigated. The theorems on the discreteness of the spectrum were proved. Also the theorems on localization and the asymptotic behavior of the eigenvalues were obtained. The properties of completeness and basicity of the system of eigenfunctions and associated (root) functions were studied. For the initial boundary value problems which generate the considered spectral problems, the theorems about existence of strong solutions were proved. Based on the already considered problems the mixed spectral conjugation problems in the case of two domains were considered. In this paper we study the perturbed initial boundary value problems generated by the sesquilinear form. In this case, the principle of superposition is used. The principle makes it possible to represent the solution of the original problem as a sum of solutions to auxiliary problems. These auxiliary problems contain inhomogeneity in only one place, that is, either in the equation or in one of the boundary conditions. The operator methods of mathematical physics in areas with Lipschitz boundaries during research are used. The original initial-boundary value problem is reduced to the Cauchy problems for a first-order integro-differential and differential-operator equations. The theorems on the existence and uniqueness of a strong solutions to the Cauchy problems in the case of two domains were proved.