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

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 $$(u_{\lambda }{)}_t=\mathrm{\Delta }{u}_{\lambda }-\mathrm{\nabla }\cdot (u_{\lambda }\chi (v_{\lambda })\mathrm{\nabla }{u}_{\lambda }),\lambda (v_{\lambda }{)}_t=\mathrm{\Delta }{v}_{\lambda }-v_{\lambda }+u_{\lambda }in\mathrm{\Omega }\times (0,\mathrm{\infty })$$ to that for the parabolic-elliptic chemotaxis system $$u_t=\mathrm{\Delta }u-\mathrm{\nabla }\cdot (u\chi (v)\mathrm{\nabla }v),0=\mathrm{\Delta }v-v+uin\mathrm{\Omega }\times (0,\mathrm{\infty }),$$ where $\mathrm{\Omega }$ 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 $$\chi (v)=\frac{{\chi }_0}{(1+v{)}^k}({\chi }_0>0,k>1)\mathrm{.}$$ 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.