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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:266

Article

Fuji, Kentarou¹  Senba, Takasi² 

[1] Tokyo Univ Sci, Tokyo 1628601, Japan

[2] Fukuoka Univ, Fukuoka, Fukuoka 8140180, Japan

关键词: Blow-up;    Chemotaxis;    Lyapunov functional;    Indirect process;   

DOI  :  10.1016/j.jde.2018.07.068

来源: Elsevier

【 摘 要 】

This paper deals with positive solutions of the fully parabolic system {u(t) = Delta u - chi(del) . (u del v) in Omega x (0, infinity), tau(1)v(t) = Delta v - v + w in Omega x (0, infinity), tau(2)w(t) = Delta w - w + u in Omega x (0,infinity), under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Omega subset of R-4 with positive parameters tau(1),tau(2), x > 0 and nonnegative smooth initial data (u(0), v(0), w(0)). Global existence and boundedness of solutions were shown if parallel to u(0)parallel to(L1(Omega)) < (8 pi)(2) /chi in Fujie-Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying parallel to u(0)parallel to(L1(Omega)) > (8 pi)(2) /chi. This result suggests that the system can be regard as a generalization of the Keller-Segel system, which has 8 pi/chi -dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R-4 . (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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