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.
ASJC Scopus subject areas
- Applied Mathematics