Stationary solutions to a Keller-Segel chemotaxis system
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Date
2006
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IOS PRESS
Abstract
We consider the following stationary Keller-Segel system from chemotaxis
Delta u - au + u(p) = 0, u > 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega,
where Omega subset of R-2 is a smooth and bounded domain. We show that given any two positive integers K,L, for p sufficiently large, there exists a solution concentrating in K interior points and L boundary points. The location of the blow-up points is related to the Green function. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.
Delta u - au + u(p) = 0, u > 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega,
where Omega subset of R-2 is a smooth and bounded domain. We show that given any two positive integers K,L, for p sufficiently large, there exists a solution concentrating in K interior points and L boundary points. The location of the blow-up points is related to the Green function. The solutions are obtained as critical points of some finite-dimensional reduced energy functional. No assumption on the symmetry, geometry nor topology of the domain is needed.
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Keywords
2-DIMENSIONAL ELLIPTIC PROBLEM, SINGULAR LIMITS, CONCENTRATING SOLUTIONS, MULTIPEAK SOLUTIONS, GLOBAL EXISTENCE, POINT DYNAMICS, UP SOLUTIONS, BLOW-UP, NEUMANN, MODEL