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

Abstract

This paper gives a first 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 $$ (u_λ)_t = Δu_λ- \nabla \cdot (u_λχ(v_λ)\nabla u_λ), \quad λ(v_λ)_t = Δv_λ- v_λ+u_λ\quad \mbox{in} \ Ω\times (0,\infty) $$ to that for the parabolic-elliptic chemotaxis system $$ u_t = Δu -\nabla \cdot (uχ(v)\nabla v), \quad 0= Δv -v +u \quad \mbox{in} \ Ω\times (0,\infty), $$ where $Ω$ is a bounded domain in $\mathbb{R}^n$ ($n\in\mathbb{N}$) with smooth boundary, $λ>0$ is a constant and $χ$ is a function generalizing $$ χ(v) = \frac{χ_0}{(1+v)^k} \quad (χ_0>0,\ k>1).$$ In chemotaxis systems parabolic-elliptic systems often provided 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. Namely, 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 $λ\searrow 0$? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.