Global well‐posedness and uniform boundedness of a higher dimensional crime model with a logistic source term. (17th January 2022)

• Record Type: Journal Article
• Authors: Wang, Deqi; Feng, Yani
• Is Part Of: Mathematical methods in the applied sciences. Volume 45:Number 8(2022)
• Page Start: 4727
• Page End: 4740
• Publication Date: 2022-01-17
• Subjects: crime model -- energy functional -- global uniform boundedness -- integral inequality; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.8066
• Languages: English
• ISSNs: 0170-4214

Abstract

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.