Existence and regularity in inverse source problem for fractional reaction-subdiffusion equation perturbed by locally Lipschitz sources

In this paper, we consider an inverse problem of determining a space-dependent source in the time fractional reaction-subdiffusion equation involving locally Lipschitz perturbations, where the additional measurements take place at the terminal time which are allowed to be nonlinearly dependent on the state. By providing regularity estimates on both time and space of resolvent operator and using local estimates on Hilbert scales, we establish some results on the existence and uniqueness of solutions and the Lipschitz type stability of solution map of the problem under consideration. In addition, when the input data take more regular values, we obtain results on regularity in time of solution for both the direct linear problem and the inverse problem above.