Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity

This paper presents global existence and boundedness of classical solutions to the fully parabolic chemotaxis system ut=Δu−∇⋅(uχ(v)∇v),vt=Δv−v+u with the strongly singular sensitivity function χ(v) such that 00,k>1). As to the regular case 00,χ0>0,k>1), it has been shown, by Winkler (2010),...

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Bibliographic Details
Published in:Applied mathematics letters 2014-12, Vol.38, p.140-143
Main Authors: Fujie, Kentarou, Yokota, Tomomi
Format: Article
Language:English
Subjects:
Boundedness
Chemotaxis
Global existence
Singular sensitivity
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Summary:This paper presents global existence and boundedness of classical solutions to the fully parabolic chemotaxis system ut=Δu−∇⋅(uχ(v)∇v),vt=Δv−v+u with the strongly singular sensitivity function χ(v) such that 00,k>1). As to the regular case 00,χ0>0,k>1), it has been shown, by Winkler (2010), that the system has a unique global classical solution which is bounded in time, whereas this method cannot be directly applied to the singular case. In the present work, a uniform-in-time lower bound for v is established and builds a bridge between the regular case as in Winkler (2010) and the singular one.
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2014.07.021