Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) in Ω × ( 0 , ∞ ) , τ 1 v t = Δ v − v + w in Ω × ( 0 , ∞ ) , τ 2 w t = Δ w − w + u in Ω × ( 0 , ∞ ) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω ⊂ R 4 with positive parameters τ 1 , τ 2 , χ > 0 and nonnegative smooth initial data ( u 0 , v 0 , w 0 ) . Global existence and boundedness of solutions were shown if ‖ u 0 ‖ L 1 ( Ω ) ( 8 π ) 2 / χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖ u 0 ‖ L 1 ( Ω ) > ( 8 π ) 2 / χ . This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8 π / χ -dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R 4 .