Global well‐posedness and uniform boundedness of a higher dimensional crime model with a logistic source term. (17th January 2022) - Wang, Deqi; Feng, Yani
Abstract : We study a class of reaction–advection–diffusion system u t = ∇ · ( ∇ u − χ ∇ v v u ) − u v + u − u 2 + β + B 1, x ∈ Ω, t > 0, v t = Δ v − v + u v + B 2, x ∈ Ω, t > 0, for χ > 0 in a smooth bounded domain Ω ⊂ ℝ n with n ≥ 2. In comparison with original urban crime models (Short et al., 2008), the system introduces the logistic source term u − u 2 + β to represent the fierce competition among criminals, which helps to reduce difficulties brought by the nonlinear growth term u v and large advection rate. We prove that, for n ≥ 2, suppose β > 0, if n < 4, n 4 − 1, if n ≥ 4, then the classical solutions ( u, v ) of the above system are uniformly bounded for any χ > 0. Our result expands χ to be arbitrary positive number.