tailieunhanh - On the solvability of the initial boundary value problem for Schrodinger systems in conical domains

In this paper, we consider the initial boundary value problem for Schrodinger systems in the cylinders with base containing the conical point. The existence and the uniqueness of the generanized solution of this problem are given. | JOURNAL OF SCIENCE OF HNUE Natural Sci. 2010 Vol. 55 No. 6 pp. 82-89 ON THE SOLVABILITY OF THE INITIAL BOUNDARY VALUE PROBLEM FOR SCHRODINGER SYSTEMS IN CONICAL DOMAINS Nguyen Thi Lien Hanoi National University of Education E-mail Lienhnue@ Abstract. In this paper we consider the initial boundary value problem for Schr odinger systems in the cylinders with base containing the conical point. The existence and the uniqueness of the generanized solution of this problem are given. Keywords Initial boundary value problem generalized solution cylinders with conical base. 1. Introduction The initial boundary value problems for Schr odinger equations in the cylinders with base containing conical points were established in 2 3 . Such problems for parabolic systems have been studied in Sobolev spaces with weights 4 5 . We are concerned with initial boundary value problems for Schr odinger sys- tems in cylinders with base containing conical point. The paper is organized is the following way. In Section 2 we define the prob- lem. In Section 3 we establish the unique existence of the generalized solution of the problem. Finally in Section 4 we apply the obtained results to a problem of mathematical physics. 2. Notations and formulation of the problem Let Ω be a bounded domain in Rn n 2 with the boundary Ω. We suppose that S Ω 0 is a smooth manifold and Ω is in a neighbourhood U of the origin 0 coincides with the cone K x x x G where G is a smooth domain on the unit sphere S n 1 in Rn . Let T be a positive real number or T . Set Ωt Ω 0 t St S 0 t . For each multi-index α α1 . . . αn Nn α α1 αn the symbol D α xα1 1 . xαnn denotes the generalized derivative of order α with respect α to x x1 . xn k tk is the generalized derivative of order k with respect to t. Let u u1 . us be a complex-valued vector function defined on ΩT . We use notation D α u D α u1 . D α us utj k u tk j u1 tj . j us tj . Let us introduce some functional spaces used in this paper see 1 82 On the .