Unbounded mass radial solutions for the Keller-Segel equation in the disk
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Date
2021
Journal Title
Journal ISSN
Volume Title
Publisher
SPRINGER HEIDELBERG
Abstract
We consider the boundary value problem
{-Delta u + u - lambda e(u) = 0 ,u > 0 in B-1(0)
partial derivative(nu)u = 0 on partial derivative B-1(0),
whose solutions correspond to steady states of the Keller-Segel system for chemotaxis. Here B-1(0) is the unit disk,. the outer normal to partial derivative B-1(0), and lambda > 0 is a parameter. We show that, provided lambda is sufficiently small, there exists a family of radial solutions u(lambda) to this system which blow up at the origin and concentrate on partial derivative B-1(0), as lambda -> 0. These solutions satisfy
lim(lambda -> 0) u lambda(0)/vertical bar in lambda vertical bar = 0 and 0 lim(lambda -> 0) 1/vertical bar in lambda vertical bar integral(B1(0)) (lambda eu lambda(x)) dx < infinity,
having in particular unbounded mass, as lambda -> 0.
{-Delta u + u - lambda e(u) = 0 ,u > 0 in B-1(0)
partial derivative(nu)u = 0 on partial derivative B-1(0),
whose solutions correspond to steady states of the Keller-Segel system for chemotaxis. Here B-1(0) is the unit disk,. the outer normal to partial derivative B-1(0), and lambda > 0 is a parameter. We show that, provided lambda is sufficiently small, there exists a family of radial solutions u(lambda) to this system which blow up at the origin and concentrate on partial derivative B-1(0), as lambda -> 0. These solutions satisfy
lim(lambda -> 0) u lambda(0)/vertical bar in lambda vertical bar = 0 and 0 lim(lambda -> 0) 1/vertical bar in lambda vertical bar integral(B1(0)) (lambda eu lambda(x)) dx < infinity,
having in particular unbounded mass, as lambda -> 0.
Description
Keywords
STATIONARY SOLUTIONS, STEADY-STATES, SYSTEM