The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity

This paper gives an insight into making a mathematical bridge between the parabolic†parabolic signal†dependent chemotaxis system and its parabolic†elliptic version. To be more precise, this paper deals with convergence of a solution for the parabolic†parabolic chemotaxis system with strong signal sensitivity (uλ)t=Δuλ−∇·(uλχ(vλ)∇vλ),λ(vλ)t=Δvλ−vλ+uλinΩ×(0,∞)to that for the parabolic†elliptic chemotaxis system ut=Δu−∇·(uχ(v)∇v),0=Δv−v+uinΩ×(0,∞),where Ω is a bounded domain in Rn (n∈N) with smooth boundary, λ>0 is a constant and χ is a function generalizing χ(v)=χ0(1+v)k(χ0>0,k>1).In chemotaxis systems parabolic†elliptic systems often gave some guide to methods and results for parabolic†parabolic systems. However, the relation between parabolic†elliptic systems and parabolic†parabolic systems has not been studied except for the case that Ω=Rn. Namely, in the case that Ω is a bounded domain, it still remains to analyze on the following question: Does a solution of the parabolic†parabolic system converge to that of the parabolic†elliptic system as λ↘0? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.