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

Abstract

This paper deals with positive solutions of the fully parabolic system{ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω⊂R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).Global existence and boundedness of solutions were shown if ‖u0‖L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖u0‖L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.