Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities

This work deals with the null controllability of an initial boundary value problem for a parabolic-elliptic coupled system with nonlinear terms of local and nonlocal kinds. The control is distributed, locally in space and appears only in one PDE. We first prove that, if the initial data is sufficiently small and the linearized system at zero satisfies an appropriate condition, the equations can be driven to zero.