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 $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ ($n\in \mathbb{N}$) with smooth boundary, $\lambda >0$ is a constant and $\chi$ is a function generalizing $$\chi (v)=\frac{{\chi }_0}{(1+v{)}^k}({\chi }_0>0,k>1).$$ 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 $\lambda \searrow 0$? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.

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