The nonlocal bistable equation: Stationary solutions on a bounded interval

We discuss instability and existence issues for the nonlocal bistable equation. This model arises as the Euler-Lagrange equation of a nonlocal, van der Waals type functional. Taking the viewpoint of the calculus of variations, we prove that for a class of nonlocalities this functional does not admit...

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Bibliographic Details
Published in:Electronic journal of differential equations 2002-01, Vol.2002 (2), p.1-12
Main Authors: Adam J. J. Chmaj, Xiaofeng Ren
Format: Article
Language:English
Subjects:
local minimizers
monotone solutions
Online Access:Get full text
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Summary:We discuss instability and existence issues for the nonlocal bistable equation. This model arises as the Euler-Lagrange equation of a nonlocal, van der Waals type functional. Taking the viewpoint of the calculus of variations, we prove that for a class of nonlocalities this functional does not admit nonconstant $C^1$ local minimizers. By taking variations along non-smooth paths, we give examples of nonlocalities for which the functional does not admit local minimizers having a finite number of discontinuities. We also construct monotone solutions and give a criterion for nonexistence of nonconstant solutions.
ISSN:1072-6691