Global solution of the chemotaxis attraction–repulsion Cauchy problem with the nonlinear signal production in R^N

In this paper, we consider the following attraction–repulsion chemotaxis model with a nonlinear signal term: u_t=∇⋅(∇u−ξ₁u∇v+ξ₂u∇w),x∈R^N,t>0,0=Δv−λ₁v+f₁(u),x∈R^N,t>0,0=Δw−λ₂w+f₂(u),x∈R^N,t>0, where ξ₁,ξ₂,λ₁,λ₂ are some positive constants, and f₁∈C¹([0,∞))satisfying0⩽f₁(s)⩽c₁s^l,∀s⩾0andl>0, f₂∈C¹([0,∞))satisfying0⩽f₂(s)⩽c₂s^m,∀s⩾0andm>0. We prove that the problem has a unique global solution.