The critical mass curve and chemotactic collapse of a two-species chemotaxis system with two chemicals.

This paper considers the following two-species chemotaxis system with two chemicals u t = Δ u − ∇ ⋅ (u ∇ v) , x ∈ Ω , t > 0 , 0 = Δ v − α v + w , x ∈ Ω , t > 0 , w t = Δ w − ∇ ⋅ (w ∇ z) , x ∈ Ω , t > 0 , 0 = Δ z − γ z + u , x ∈ Ω , t > 0 subject to the homogeneous Neumann boundary condition with α , γ > 0 , where Ω ⊂ R 2 is a smooth bounded domain. In the previous paper [Yu et al, Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals, Nonlinearity 31 (2018) 502–514], we proved that the system possesses finite-time blow-up solutions provided (0.1) m u m w − 2 π (m u + m w) > 0 , where m u and m w denote the initial masses of u and w respectively. In this paper, we first establish the critical mass curve m u m w − 2 π (m u + m w) = 0 by proving that solutions are globally bounded whenever (0.2) m u m w − 2 π (m u + m w) < 0 , which means the blow-up condition (0.1) is optimal. Based on this, we further show for the blow-up region (0.1) that (u (x , t) d x , w (x , t) d x) forms a delta function singularity at the blow-up point x 0 as t → T max with the collapse mass (M x 0 u , M x 0 w) satisfying M x 0 u M x 0 w − M ∗ (M x 0 u + M x 0 w) ≥ 0 , where M ∗ = 4 π for x 0 ∈ Ω and M ∗ = 2 π for x 0 ∈ ∂ Ω. [ABSTRACT FROM AUTHOR]