Solvability of Boundary Value Problems for Degenerate Equations of~Sobolev Type

N. R. Pinigina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 120-129

Résumé

The aim of this work is to prove the existence and uniqueness of regular solutions of the first boundary value problem for the systems of Sobolev type equations with elliptic-parabolic operators with spatial degeneracy. By A. I. Kozhanov considered the initial-boundary value problems for Sobolev type equations with elliptic-parabolic operators of the second order acting on the space variables. The existence of solutions under the conditions «characteristic bulge» of the border area with respect to the spatial operators have been proved in the works. The technique used in this paper will be close to the technique of above author. For the study of degenerate systems of Sobolev type equations used the combination of the regularization method and the method of a priori estimates. It is constructed a family of approximate solutions of degenerate equations by the regularization method. Analysis of integral inequalities in obtaining of priori estimates, based on the integration by parts and in using of Cauchy–Bunyakovskii, Hölder's and Young's inequalities. The properties of weighted Sobolev spaces also ate used.

Détail

Keywords: the boundary value problem, the Sobolev type equation, regular solutions, a priori estimates.