Explicit lower bound of blow–up time for an attraction–repulsion chemotaxis system

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)inΩ×(0,t⁎),0=Δv+αu−βvinΩ×(0,t⁎),0=Δw+γu−δwinΩ×(0,t⁎), where Ω is a smooth and bounded domain of R2, t⁎ is the blow–up time and α,β,γ,δ,χ,ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u(x,0)=u0∈C0(Ω¯), system (◇) has a unique classical solution which becomes unbounded as t↗t⁎. The main result of this investigation is to provide an explicit lower bound for t⁎ estimated in terms of ∫Ωu02dx and attained by means of well–established techniques based on ordinary differential inequalities.