Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production 𝜕tu=Δu−∇·χ1u∇v1+∇·(χ2u∇v2),x∈R2,t>0,0=Δvj−λjvj+w,x∈R2,t>0,(j=1,2),𝜕tw+δw=u,x∈R2,t>0,u(0,x)=u0(x),w(0,x)=w0(x),x∈R2, where the parameters χi≥0, λi>0(i=1,2) and non-negative initial data (u0(x),w0(x))∈L1(R2)∩L∞(R2). We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of χ1≥χ2. At the same time, we propose that when the radial solution satisfies χ1−χ2≤2πδ∥u0∥L1(R2)+∥w0∥L1(R2), the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution.