A refined criterion and lower bounds for the blow-up time in a parabolic–elliptic chemotaxis system with nonlinear diffusion

Abstract

This paper deals with unbounded solutions to the following zero-flux chemotaxis system (♢)ut=∇⋅[(u+α)m1−1∇u−χu(u+α)m2−2∇v](x,t)∈Ω×(0,Tmax),0=Δv−M+u(x,t)∈Ω×(0,Tmax),where α>0, Ω is a smooth and bounded domain of Rn, with n≥1, t∈(0,Tmax) is Tmax the blow-up time, and m1,m2 are real numbers. Given a sufficiently smooth initial data u0≔u(x,0)≥0 and set M≔1|Ω|∫Ωu0(x)dx, from the literature it is known that under a proper interplay between the above parameters m1,m2 and the extra condition ∫Ωv(x,t)dx=0, system (♢) possesses for any χ>0 a unique classical solution which becomes unbounded at t↗Tmax. In this investigation we first show that for p0>n2(m2−m1) any blowing up classical solution in L∞(Ω)-norm blows up also in Lp0(Ω)-norm. Then we estimate the blow-up time Tmax providing a lower bound T.