Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains

In this paper, we establish a local null controllability result for a nonlinear parabolic PDE with local and nonlocal nonlinearities in a domain whose boundary moves in time by a control force with a multiplicative part acting on a prescribed subdomain. We prove that, if the initial data is sufficiently small and the linearized system at zero satisfies an appropriate condition, the equation can be driven to zero.