The fast signal diffusion limit in a Keller-Segel system

This paper deals with convergence of a solution for the parabolic-parabolic Keller-Segel system $$(u_\lambda)_t = \Delta u_\lambda - \chi \nabla \cdot (u_\lambda \nabla v_\lambda), \quad \lambda (v_\lambda)_t = \Delta v_\lambda - v_\lambda + u_\lambda \quad \mbox{in} \ \Omega\times (0,\infty)$$ to that for the parabolic-elliptic Keller-Segel system $$u_t = \Delta u - \chi \nabla \cdot (u \nabla v), \quad 0= \Delta v -v +u \quad \mbox{in} \ \Omega\times (0,\infty)$$ as $\lambda \searrow 0$, where $\Omega$ is a bounded domain in $\mathbb{R}^n$ ($n\ge 2$) with smooth boundary, $\chi, \lambda>0$ are constants. In chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, there have not been rich results on the relation between parabolic-elliptic systems and parabolic-parabolic systems. Namely, it still remains to analyze on the following question except some cases: Does a solution of the parabolic-parabolic system converge to that of the parabolic-elliptic system as $\lambda \searrow 0$? In the case that $\Omega$ is the whole space $\mathbb{R}^n$, or $\Omega$ is a bounded domain and $\chi$ is a strong signal sensitivity, some positive answers were shown in the previous works. Therefore, one can expect a positive answer to this question also in the Keller-Segel system in a bounded domain $\Omega$ in some cases. This paper gives some positive answer in the 2-dimensional and the higher-dimensional Keller-Segel system.