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

• Main Authors: Chunlong Sun, Qian Liu, Gongsheng Li
• Format: Article
• Language: English
• Published: Hindawi Limited 2017-01-01
• Series: Advances in Mathematical Physics
• Online Access: http://dx.doi.org/10.1155/2017/6801260

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.