Application of an Adams type inequality to a two-chemical substances chemotaxis system

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 homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive...

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Bibliographic Details
Published in:Journal of Differential Equations 2017-07, Vol.263 (1), p.88-148
Main Authors: Fujie, Kentarou, Senba, Takasi
Format: Article
Language:English
Subjects:
Adams' inequality
Chemotaxis
Global existence
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 homogeneous Neumann boundary conditions or mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded domain Ω⊂Rn (n≤4) with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0). In the lower dimensional case (n≤3), it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded. In the case n=4, it is shown that in the radially symmetric setting solutions to the Neumann boundary value problem of the system exist globally in time and remain bounded if ‖u0‖L1(Ω)
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2017.02.031