The Nonlocal Bistable Equation: Stationary Solutions on a Bounded Interval
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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.
CitationChmaj, A. J. J., & Ren, X. (2002). The nonlocal bistable equation: Stationary solutions on a bounded interval. Electronic Journal of Differential Equations, 2002(02), pp. 1-12.
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