Boundedness and asymptotic behavior of solutions to one-dimensional urban crime system with nonlinear diffusion

Abstract

This paper deals with a one-dimensional cross-diffusion system ut=(D(u)ux)x−χ(uvvx)x−uv+B1(x,t),x∈Ω,t>0,vt=vxx−v+uv+B2(x,t),x∈Ω,t>0,which is proposed by Short et al. (2008) to describe the dynamics of urban crime. If D(u)≥D0(u+1)m−1 with D0,m>0, it is proved for arbitrary χ>0 that the system possesses a globally bounded classical solution provided m>14 with some mild assumptions on nonnegative functions B1,B2. In addition, if B2≡0, the attractiveness value of v and its derivative vx decay to zero in the long time limit.