Inverse problem of finding n coefficients of lower derivatives in a parabolic equation

We consider the problem of reconstructing the vector function $\vec b(x) = (b_1 ,...,b_n )$ in the term $(\vec b,\nabla u)$ in a linear parabolic equation. This coefficient inverse problem is considered in a bounded domain Ω ? R ⁿ . To find the above-mentioned function $\vec b(x)$ , in addition to initial and boundary conditions we pose an integral observation of the form $\int_0^T {u(x,t)\vec \omega (t)dt = \vec \chi (x)} $ , where $\vec \omega (t) = (\omega _1 (t),...,\omega _n (t))$ is a given weight vector function. We derive sufficient existence and uniqueness conditions for the generalized solution of the inverse problem. We present an example of input data for which the assumptions of the theorems proved in the paper are necessarily satisfied.