Problems of determining the unknown source in parabolic and hyperbolic equations

Initial-boundary value problems for a parabolic and a hyperbolic equation with a source are considered. The hyperbolic equation involves the second time derivative multiplied by a positive parameter $\varepsilon$ and coincides with the parabolic equation when $\varepsilon$ is zero. The source is the sum of two unknown functions of spatial variables multiplied by exponentially decaying functions of time. The inverse problems of determining the unknown functions of spatial variable from additional solution information that represents a function of time are considered. It is proved that the inverse problem for the parabolic equation has an infinite set of solutions, while, for any positive $\varepsilon$, the inverse problem for the hyperbolic equation has a unique solution.