Large mass boundary condensation patterns in the stationary Keller–Segel system

Abstract

We consider the boundary value problem

{ -Delta u + u = lambda e(u), in Omega partial derivative(v)u = 0 on partial derivative Omega

where Omega is a bounded smooth domain in R-2, lambda > 0 and v is the inner normal derivative at partial derivative Omega. This problem is equivalent to the stationary Keller-Segel system from chemotaxis.

We establish the existence of a solution u(lambda) which exhibits a sharp boundary layer along the entire boundary partial derivative Omega as lambda -> 0. These solutions have large mass in the sense that integral(Omega) lambda e(u lambda) similar to | log lambda|. (C) 2016 Elsevier Inc. All rights reserved.