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

Let Q be a heat conduction body and let ϕ = ϕ ( t ) be given. We consider the problem of finding a two-dimensional heat source having the form ϕ ( t ) f ( x , y ) in Q . The problem is ill-posed. Assuming ∂ Q is insulated and ϕ ≢ 0 , we show that the heat source is defined uniquely by the temperature history on ∂ 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.