Global well-posedness and uniform boundedness of 2D urban crime models with nonlinear advection enhancement

Abstract We study the global well-posedness and uniform boundedness of a two-dimensional reaction–advection–diffusion system with nonlinear advection. This strongly coupled system of nonlinear partial differential equations represents the continuum of a 2D lattice model designed to describe residential burglary, where each location is characterised by a tractability value that varies in both space and time. We show that the model with sublinear advection enhancement is globally well-posed, with a unique solution that is classical and uniformly bounded in time. Our results provide valuable insights into the development of urban crime models with nonlinear advection enhancements, making them suitable for broader applications, including nonlocal or heterogeneous near-repeat victimisation effects.