Large time behavior of solutions for the attraction–repulsion Keller–Segel system with large initial data.

In this paper, we study the following attraction–repulsion Keller–Segel system u t = Δ u − ∇ ⋅ ( χ u ∇ v ) + ∇ ⋅ ( ξ u ∇ w ) , x ∈ Ω , t > 0 , v t = Δ v + α u − β v , x ∈ Ω , t > 0 , 0 = Δ w + γ u − δ w , x ∈ Ω , t > 0 , ∂ u ∂ ν = ∂ v ∂ ν = ∂ w ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , in a bounded domain Ω ⊂ R 2 with smooth boundary. The boundedness of solutions with arbitrarily large initial data has been proved in the case of ξ γ ≥ χ α (Jin and Wang, 2016). Under the additional assumption ξ γ β ≥ χ α δ , we show that the global classical solution will converge to the unique constant state ( u ̄ 0 , α β u ̄ 0 , γ δ u ̄ 0 ) as t → + ∞ . [ABSTRACT FROM AUTHOR]