Determination of a two-dimensional heat source: Uniqueness, regularization and error estimate

Let $Q$ be a heat conduction body and let $\varphi = \varphi(t)$ be given. We consider the problem of finding a two-dimensional heat source having the form $\varphi(t)f(x,y)$ in $Q$. The problem is ill-posed. Assuming $\partial Q$ is insulated and $\varphi \not\equiv 0$, we show that the heat source is defined uniquely by the temperature history on $\partial Q$ and the temperature distribution in $Q$ at the initial time $t = 0$ and at the final time $t = 1$. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.