Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species

In this paper, we consider the fully parabolic chemotaxis system for two species {∂tu1=Δu1−χ1∇⋅(u1∇v),x∈Ω,t>0,∂tu2=Δu2−χ2∇⋅(u2∇v),x∈Ω,t>0,∂tv=Δv−γv+α1u1+α2u2,x∈Ω,t>0, with homogeneous Neumann boundary condition in the dimension n≥3n≥3, where ΩΩ is a ball in RnRn and χ1,χ2,γ,α1,α1χ1,χ2,γ,α1,α1 are positive constants. We consider the more general case χ1≠χ2χ1≠χ2. It is proved that for any mi>0,(i=1,2)mi>0,(i=1,2), there exists radially symmetric initial data (u10,u20,v0)∈(C0(Ω̄))2×W1,∞(Ω) with mi=∫Ωui0(i=1,2) such that the corresponding solution blows up in finite time in the sense limt→T‖u1‖L∞(Ω)+‖u2‖L∞(Ω)=∞limt→T‖u1‖L∞(Ω)+‖u2‖L∞(Ω)=∞ for some 0