The fast signal diffusion limit in a Keller-Segel system

Abstract

This paper deals with convergence of a solution for the parabolic-parabolic Keller-Segel system \[ (u_λ)_t = Δu_λ- χ\nabla \cdot (u_λ\nabla v_λ), \quad λ(v_λ)_t = Δv_λ- v_λ+ u_λ\quad \mbox{in} \ Ω\times (0,\infty) \] to that for the parabolic-elliptic Keller-Segel system \[ u_t = Δu - χ\nabla \cdot (u \nabla v), \quad 0= Δv -v +u \quad \mbox{in} \ Ω\times (0,\infty) \] 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.

arXiv comment20 pages. arXiv admin note: text overlap with arXiv:1711.01677