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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 6801260, 6 pages
https://doi.org/10.1155/2017/6801260
Research Article

Conditional Well-Posedness for an Inverse Source Problem in the Diffusion Equation Using the Variational Adjoint Method

1School of Science, Shandong University of Technology, Zibo 255049, China
2Department of Mathematics, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Gongsheng Li

Received 22 March 2017; Accepted 31 May 2017; Published 27 June 2017

Academic Editor: Mikhail Panfilov

Copyright © 2017 Chunlong Sun et al. 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

This article deals with an inverse problem of determining a linear source term in the multidimensional diffusion equation using the variational adjoint method. A variational identity connecting the known data with the unknown is established based on an adjoint problem, and a conditional uniqueness for the inverse source problem is proved by the approximate controllability to the adjoint problem under the condition that the unknowns can keep orders locally. Furthermore, a bilinear form is set forth also based on the variational identity and then a norm for the unknowns is well-defined by which a conditional Lipschitz stability is established.