The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coecient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the xed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.