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
UR - http://www.scopus.com/inward/citedby.url?scp=84907056747&partnerID=8YFLogxK
U2 - 10.1016/j.aml.2014.07.021
DO - 10.1016/j.aml.2014.07.021
M3 - Article
AN - SCOPUS:84907056747
SN - 0893-9659
VL - 38
SP - 140
EP - 143
JO - Applied Mathematics Letters
JF - Applied Mathematics Letters
ER -