The critical mass curve and chemotactic collapse of a two-species chemotaxis system with two chemicals

Abstract

This paper considers the following two-species chemotaxis system with two chemicals ut=Δu−∇⋅(u∇v),x∈Ω,t>0,0=Δv−αv+w,x∈Ω,t>0,wt=Δw−∇⋅(w∇z),x∈Ω,t>0,0=Δz−γz+u,x∈Ω,t>0subject to the homogeneous Neumann boundary condition with α,γ>0, where Ω⊂R2 is a smooth bounded domain. In the previous paper [Yu et al, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity 31 (2018) 502–514], we proved that the system possesses finite-time blow-up solutions provided (0.1)mumw−2π(mu+mw)>0,where mu and mw denote the initial masses of u and w respectively. In this paper, we first establish the critical mass curve mumw−2π(mu+mw)=0 by proving that solutions are globally bounded whenever (0.2)mumw−2π(mu+mw)<0,which means the blow-up condition (0.1) is optimal. Based on this, we further show for the blow-up region (0.1) that (u(x,t)dx,w(x,t)dx) forms a delta function singularity at the blow-up point x0 as t→Tmax with the collapse mass (Mx0u,Mx0w) satisfying Mx0uMx0w−M∗(Mx0u+Mx0w)≥0,where M∗=4π for x0∈Ω and M∗=2π for x0∈∂Ω.