An inverse source problem for parabolic equations with local measurements

Inverse source problems for partial differential equations are of interest both in mathematics and in engineering. In this paper, we consider the problem of recovering the spatial contamination source in the diffusion model with the local measurements at fixed times. One example is constructed to show that the local measurements at one fixed time are not enough to determine the source. We propose the problem of recovering the source with the local measurements at two different fixed times. The uniqueness and conditional stability estimate are proved by the analytic continuation method.