Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time

The inverse problem of finding the coefficient $\rho(x)=\rho_0+r(x)$ multiplying $u_t$ in the heat equation is studied. The unknown function $r(x)\geqslant0$ is sought in the class of bounded functions, and $\rho_0$ is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form $\int\limits_0^T u(x,t)d\mu(t)=\chi(x)$ with a given measure $d\mu(t)$ and a function $\chi(x)$. The case of integral observation (i.e., $d\mu(t)=\omega(t)dt$) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy-to-check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.