Boundedness, blowup and critical mass phenomenon in competing chemotaxis

Abstract We consider 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 ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ Ω , with homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R 2 with smooth boundary. The system models the chemotactic interactions between one species (denoted by u ) and two competing chemicals (denoted by v and w ), which has important applications in Alzheimer's disease. Here all parameters χ , ξ , α , β , γ and δ are positive. By constructing a Lyapunov functional, we establish the global existence of uniformly-in-time bounded classical solutions with large initial data if the repulsion dominates or cancels attraction (i.e., ξ γ ≥ α χ ). If the attraction dominates (i.e., ξ γ α χ ), a critical mass phenomenon is found. Specifically speaking, we find a critical mass m ⁎ = 4 π α χ − ξ γ such that the solution exists globally with uniform-in-time bound if M m ⁎ and blows up if M > m ⁎ and M ∉ { 4 π m θ : m ∈ N + } where N + denotes the set of positive integers and M = ∫ Ω u 0 d x the initial cell mass.