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{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 smo...

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Bibliographic Details
Published in:Journal of Differential Equations 2019-01, Vol.266 (2-3), p.942-976
Main Authors: Fujie, Kentarou, Senba, Takasi
Format: Article
Language:English
Subjects:
Blow-up
Chemotaxis
Indirect process
Lyapunov functional
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Summary: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. 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.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2018.07.068