  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 $$ (u_\lambda)_t = \Delta u_\lambda - \nabla \cdot
(u_\lambda \chi(v_\lambda)\nabla u_\lambda), \quad \lambda (v_\lambda)_t =
\Delta v_\lambda - v_\lambda +u_\lambda \quad \mbox{in} \ \Omega\times
(0,\infty) $$ to that for the parabolic-elliptic chemotaxis system $$ u_t =
\Delta u -\nabla \cdot (u\chi(v)\nabla v), \quad 0= \Delta v -v +u \quad
\mbox{in} \ \Omega\times (0,\infty), $$ 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 $$ \chi(v) =
\frac{\chi_0}{(1+v)^k} \quad (\chi_0>0,\ k>1).$$ 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.
