Global existence in a two-dimensional nonlinear diffusion model for urban crime propagation

Abstract

In this paper, we will investigate the urban crime system with the nonlinear diffusion ∂tu=∇⋅um−1∇u−χ∇⋅uv∇v−uv+B1(x,t),x∈Ω,t>0,∂tv=Δv+uv−v+B2(x,t),x∈Ω,t>0under no-flux initial–boundary conditions in a bounded convex domain Ω⊂R2 with smooth boundary. Our first result asserts that when m>32, χ∈(0,∞) and when 1<m≤32, χ∈0,32, the nonlinear diffusion system possesses locally bounded solutions for arbitrary initial data and sufficiently regular source terms B1 and B2. Our second result further reveals that the solutions can be demonstrated to be globally bounded if the source terms satisfy mild additional conditions. In particular, this improves the related result for m>32 (Rodríguez and Winkler, 2020) to the case of arbitrary m>1 under the additional assumption on χ.