Finite-time blow-up and boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear signal productions

Abstract

This article is focused on the no-flux attraction–repulsion chemotaxis model ut=∇⋅(D(u)∇u−S(u)∇v+T(u)∇w)+h(u),x∈Ω,t>0,0=Δv−α(t)+f(u),x∈Ω,t>0,0=Δw−β(t)+g(u),x∈Ω,t>0defined in a smooth and bounded domain Ω⊂Rn(n≥2) and subjected to homogeneous Neumann boundary conditions. The functions D,S and T suitably generalize the singular prototypes D(s)=(s+1)m1−1,S(s)=s(s+1)m2−1,T(s)=s(s+1)m3−1,s≥0,m1,m2,m3∈R,and f and g extend the prototypes f(s)=(s+1)γ1,g(s)=(s+1)γ2,s≥0,γ1,γ2>0,as well as the logistic dampening satisfies h(s)=ks−μsl with k∈R, μ>0, l≥1. Under alternative conditions, we establish the global existence and boundedness of classical solutions to the above model. However, when m2=m3 and no logistic source is taken into account, the finite time blow-up phenomenon preserves provided that γ1>γ2 and γ1>m1∗−m2+2n with m1∗=max{1,m1}.