Global bounded solutions and their asymptotic properties under small initial data condition in a two-dimensional chemotaxis system for two species

In this paper, a fully parabolic chemotaxis system for two species{ut=Δu−∇⋅(u∇w),x∈Ω,t>0,vt=Δv−∇⋅(v∇w),x∈Ω,t>0,wt=Δw+u−w−vw,x∈Ω,t>0 is considered under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2. We obtain the global boundedness and asymptotic behavior with small initial data condition in critical space. More precisely, it is proved that one can find a small ε0>0 such that for any initial data (u0,v0,w0) satisfying ||u0||L1(Ω)<ε0 and ||∇w0||L2(Ω)<ε0, the solution of the problem above is global in time and bounded. In addition, (u,v,w) converges to the steady state (m1,m2,m11+m2) as t→∞, where m1:=1|Ω|∫Ωu0 and m2:=1|Ω|∫Ωv0.