Stability Of Steady State Solutions For A Class Of Keller-Segel Model

The Keller and Segel model is a computational and modelling way to describe chemotaxis.The study of the Keller and Segel model is one of the most popular research fields in biomathematics.In this paper,we mainly study a class of Keller and Segel model.Firstly,we simply introduce the background of the Keller and Segel model and recall the history of mathematical research on chemotaxis.Secondly,we show the basic profile of the spiky solution as k? ?,then use the matched asymptotic method to obtain asymptotic expansion of the spiky solution in terms of powers of k^-1.Finally,in order to prove the spiky solution is locally asymptotically stable,we investigate the eigenvalue problem corresponding to the chemotaxis model,then we only need to prove that all the eigenvalues of the eigenvalue problem are negative.