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 ( ◇ ) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) in Ω × ( 0 , t ⁎ ) , 0 = Δ v + α u − β v in Ω × ( 0 , t ⁎ ) , 0 = Δ w + γ u − δ w in Ω × ( 0 , t ⁎ ) , where Ω is a smooth and bounded domain of R 2 , 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 ) = u 0 ∈ C 0 ( 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 ∫ Ω u 0 2 d x and attained by means of well–established techniques based on ordinary differential inequalities.