Well-posedness for a model derived from an attraction–repulsion chemotaxis system

In this paper, we are interested in a model derived from an attraction–repulsion chemotaxis system in high dimensions:{∂tu−Δu=−β1∇⋅(u∇v)+β2∇⋅(u∇w),x∈Rn,t>0,λ1v−Δv=u,x∈Rn,t>0,λ2w−Δw=u,x∈Rn,t>0,u(x,0)=u0(x),x∈Rn, with the parameters β1≥0β1≥0, β2≥0β2≥0, λ1>0λ1>0, λ2>0λ2>0 and nonnegative initial data u0(x)∈L1(Rn)∩L∞(Rn)u0(x)∈L1(Rn)∩L∞(Rn). We prove that a global bounded solution exists when the repulsion prevails over the attraction in the sense of β1<β1β1<β1. Moreover, we give the smoothness of the solution and obtain its decay rates in Ws,p(Rn)Ws,p(Rn), which coincide with the ones of the classical heat equation. Conversely, when n=2n=2, β1>β2β1>β2, we prove that the finite time blow-up may occur.