The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation

We study the inverse problem for a parabolic equation of recovering the source, that is, the right-hand side $F(x,t)=h(x,t)f(x)$, where the function $f(x)$ is unknown. To find $f(x)$, along with the initial and boundary conditions, we also introduce an additional condition of nonlocal observation of the form $\displaystyle\int_{0}^{T}u(x,t)\,d\mu(t)=\chi(x)$. We prove the Fredholm property for the problem stated in this way, and obtain sufficient conditions for the existence and uniqueness of a solution. These conditions are of the form of readily verifiable inequalities and put no restrictions on the value of $T>0$ or the diameter of the domain $\Omega$ under consideration. The proof uses a priori estimates and the qualitative properties of solutions of initial-boundary value problems for parabolic equations.
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