Kentarou Fujie

Boundedness in a fully parabolic chemotaxis system with strongly singular sensitivity

Abstract 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.

Global existence and boundedness of radial solutions to a two dimensional fully parabolic chemotaxis system with general sensitivity

This paper deals with positive radially symmetric solutions of the Neumann boundary value problem for the fully parabolic chemotaxis system, {ut=Δu−∇⋅(u∇χ(v))in Ω×(0,∞),τvt=Δv−v+uin Ω×(0,∞), in a ball with general sensitivity function satisfying and decaying property (), parameter and nonnegative radially symmetric initial data. It is shown that if is sufficiently small, then the problem has a unique classical radially symmetric solution, which exists globally and remains uniformly bounded in time. Especially, this result establishes global existence of solutions in the case for all , which has been left as an open problem.