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 u t = Δ u − ∇ ⋅ ( u χ ( v ) ∇ v ) , v t = Δ v − v + u with the strongly singular sensitivity function χ ( v ) such that 0 < χ ( v ) ≤ χ 0 v k ( χ 0 > 0 , k > 1 ) . As to the regular case 0 < χ ( v ) ≤ χ 0 ( 1 + α v ) k ( α > 0 , χ 0 > 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.