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

Abstract

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.