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

This paper deals with unbounded solutions to the following zero-flux chemotaxis system ut=∇⋅[(u+α)mjavax.xml.bind.JAXBElement@2d15ccb6−1∇u−χu(u+α)mjavax.xml.bind.JAXBElement@6eb737ca−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 [Formula presented], 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 [Formula presented] any blowing up classical solution in L∞(Ω)-norm blows up also in Lpjavax.xml.bind.JAXBElement@eea994b(Ω)-norm. Then we estimate the blow-up time Tmax providing a lower bound T.