Electron. J. Diff. Eqns., Vol. 2002(2002), No. 02, pp. 1-12.

The nonlocal bistable equation: Stationary solutions on a bounded interval

Adam J. J. Chmaj & Xiaofeng Ren

Abstract:
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 C1 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.

Submitted July 18, 2001. Published January 2, 2002.
Math Subject Classifications: 45G10.
Key Words: local minimizers, monotone solutions.

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Adam J. J. Chmaj
Department of Mathematics,
Heriot-Watt University Riccarton,
Edinburgh EH14 4AS, Scotland, UK
e-mail: A.J.Chmaj@ma.hw.ac.uk

Xiaofeng Ren
Department of Mathematics and Statistics
Utah State University
Logan, UT 84322, USA
e-mail: ren@math.usu.edu

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