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Journal of Function Spaces
Volume 2016, Article ID 5687920, 15 pages
http://dx.doi.org/10.1155/2016/5687920
Research Article

Nonclassical Problem for Ultraparabolic Equation in Abstract Spaces

1Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, 3 I. Tchavtchavadze Avenue, 0179 Tbilisi, Georgia
2School of Informatics, Engineering and Mathematics, University of Georgia, 77a M. Kostava Street, 0175 Tbilisi, Georgia

Received 6 March 2016; Accepted 24 April 2016

Academic Editor: Maria Alessandra Ragusa

Copyright © 2016 Gia Avalishvili and Mariam Avalishvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Nonclassical problem for ultraparabolic equation with nonlocal initial condition with respect to one time variable is studied in abstract Hilbert spaces. We define the space of square integrable vector-functions with values in Hilbert spaces corresponding to the variational formulation of the nonlocal problem for ultraparabolic equation and prove trace theorem, which allows one to interpret initial conditions of the nonlocal problem. We obtain suitable a priori estimates and prove the existence and uniqueness of solution of the nonclassical problem and continuous dependence upon the data of the solution to the nonlocal problem. We consider an application of the obtained abstract results to nonlocal problem for ultraparabolic partial differential equation with second-order elliptic operator and obtain well-posedness result in Sobolev spaces.