TY - JOUR

T1 - Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity

AU - Fujie, Kentarou

AU - Yokota, Tomomi

N1 - Funding Information:
The authors would like to thank the anonymous referees for useful comments. This work was partially supported by Grant-in-Aid for Scientific Research (C) (No. 25400119 ), JSPS .

PY - 2014/12

Y1 - 2014/12

N2 - 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 0<χ(v)≤χ0 vk(χ0>0,k>1). As to the regular case 0<χ(v)≤χ0(1+αv)k(α>0, 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.

AB - 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 0<χ(v)≤χ0 vk(χ0>0,k>1). As to the regular case 0<χ(v)≤χ0(1+αv)k(α>0, 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.

KW - Boundedness

KW - Chemotaxis

KW - Global existence

KW - Singular sensitivity

UR - http://www.scopus.com/inward/record.url?scp=84907056747&partnerID=8YFLogxK

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U2 - 10.1016/j.aml.2014.07.021

DO - 10.1016/j.aml.2014.07.021

M3 - Article

AN - SCOPUS:84907056747

VL - 38

SP - 140

EP - 143

JO - Applied Mathematics Letters

JF - Applied Mathematics Letters

SN - 0893-9659

ER -