We consider an inverse problem of finding a solution u and a coefficient in the equation

where l ₀ is an elliptic operator of the second order and l ₁ is a first order operator. A function a(x) is unknown on some subset G ₀ ⊂ G and is given on the set G \ G ₀. The conditions of the first boundary value problem are augmented with the overdetermination condition on the set G ₀, which can be considered as the partial final overdetermination condition. Under some conditions on the data of the problem, the existence and uniqueness questions are studied.