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

Abstract

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 parabolicparabolic chemotaxis system with strong signal sensitivityto that for the parabolic-elliptic chemotaxis systemwhere Ω is a bounded domain in R n (n ∈ N) with smooth boundary, λ > 0 is a constant and χ is a function generalizingIn chemotaxis systems parabolic-elliptic systems often provided some guide to methods and results for parabolic-parabolic systems. However, the relation between parabolicelliptic 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 λ ց 0? This paper gives some positive answer in the chemotaxis system with strong signal sensitivity.2010Mathematics Subject Classification. Primary: 35A09; Secondary: 35K51, 35J15, 92C17.

The fast signal diffusion limit in a Keller–Segel system

“…In this paper our aim is, by considering that the parabolic-elliptic system is as a limit of its parabolic-parabolic case, to establish a result such that only dealing with the parabolic-parabolic Keller-Segel system is enough to obtain new properties for solutions of its parabolic-elliptic case. As a related work, in the study of a chemotaxis system with signal-dependent sensitivity, some result on this subject has already been obtained ( [18]); however, in this study we could not attain a result on a minimal Keller-Segel system from a technical reason. Thus the subject of this paper is a challenging problem for a progress of the chemotaxis system.…”

“…, a positive answer to this question is also shown under the condition that χ 0 is small [18]. Therefore we can expect a positive answer to this question also in the Keller-Segel system in a bounded domain Ω in some case.…”

“…Thus we cannot use methods only for parabolic equations and only for elliptic equations when we would like to obtain some error estimate for solutions of (1.1) and those of (1.2), and it seems to be difficult to combine these methods. Therefore we rely on a compactness method to obtain convergence of a solution (u λ , v λ ) as λ ց 0, which is the same strategy as that of the proof of [18,Theorem 1.3]. In order to use a compactness method some estimate for the solution uniformly in time and λ is required.…”

Erratum to the paper “The fast signal diffusion limit in a chemotaxis system with strong signal sensitivity”

“…We became aware of mistakes in the proof of [, Theorem 1.3]. We would like to thank Professor Michael Winkler for pointing out these mistakes, and moreover, appreciated his modifying arguments in his joint paper with Professors Yulan Wang and Zhaoyin Xiang ([]).…”

“…We would like to thank Professor Michael Winkler for pointing out these mistakes, and moreover, appreciated his modifying arguments in his joint paper with Professors Yulan Wang and Zhaoyin Xiang ([]). In the following notations of the paper are used. The estimate obtained in Theorem 1.2, is correct under the condition that .…”