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+uut=Δu−∇⋅(uχ(v)∇v),vt=Δv−v+u with the strongly   singular sensitivity function χ(v)χ(v) such that 0<χ(v)≤χ0vk(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 vv is established and builds a bridge between the regular case as in Winkler (2010) and the singular one.