Some progress for global existence and boundedness in a multi-dimensional parabolic–elliptic two-species chemotaxis system with indirect pursuit-evasion interaction

This paper deals with the Neumann problem for the two-species chemotaxis system (0.1)ut=Δu−χ∇⋅(u∇w),x∈Ω,t>0,vt=Δv+ξ∇⋅(v∇z),x∈Ω,t>0,0=Δw−w+v,x∈Ω,t>0,0=Δz−z+u,x∈Ω,t>0,in a smoothly bounded domain Ω⊂RN (N≤3) with nonnegative initial data. The parameters χ and ξ are positive. The present work asserts that this problem admits a globally defined bounded classical solution. We use a new method to obtain the L∞-bound of the solution, which is different from the method of Li et al. (2020), Liu and Zheng (2023), Zheng and Zhang (2023). This result improves or extends previous results of several authors.