In this study, we examine the inverse source problem for the Poisson equation under two specific conditions: terminal data and integral nonlocal data, which is classified as ill-posed in the Hadamard sense. The first contribution involves the regularization of the final value problem with a separable source function and noisy data in the \(L^p\) space, with \(1 < p \le 2\) , extending the typical approach that uses in \(L^2\) . We also demonstrate the convergence behavior of the source term as the parameter \(k \rightarrow 0^+\) . The second contribution presents a regularization result for the problem with a nonlocal integral condition, rather than a final condition, offering regularization techniques in both \(L^2\) and \(L^p\) spaces and demonstrating the ill-posed nature of the problem in \(L^p\) . This work is, to the authors’ knowledge, this is the first study to address the nonlocal condition within the inverse source problem for the Poisson equation.
