A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions

Abstract

In this paper we consider the quasilinear chemotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+f(u),x∈Ω,t>0,0=Δv−μ(t)+u,x∈Ω,t>0, with homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with n≥2, where χ>0, μ(t):=1|Ω|∫Ωu(x,t)dx and f∈C([0,∞))∩C1((0,∞)) is a logistic source of the form f(s)=as−bsκ with a≥0,b>0, κ>1 and s≥0, and the diffusion D∈C2([0,∞)) is supposed to satisfyD(s)≥D0s−mfor alls>0 with some D0>0 and m∈R. Given any b>0, when the logistic source is strong enough in the sense thatκ>m+3−4n+2andκ>2, it is shown that for any initial data u0∈C0(Ω¯) and n≥2 the problem possesses a unique global bounded classical solution. However, whenD(s)=D0s−mfor alls>0 with 4n−1<m≤0 in the sense that n≥5, and the effect of logistic source is weaker in the sense thatκ∈(1,(3−m)n−22n−2), it is shown that for arbitrary prescribed M0>0 there exists initial data u0∈C∞(Ω¯) satisfying ∫Ωu0=M0 such that the corresponding solution (u,v) of the system blows up in finite time in a ball Ω=B0(R)⊂Rn with some R>0. This result extends the blow-up arguments of the Keller–Segel chemotaxis model with logistic cell kinetics in Winkler [39] to more general quasilinear case. Moreover, since there is a gap in the proof of Zheng et al. [46], it also presents modified results for the mistake.