The Nonlocal Bistable Equation: Stationary Solutions on a Bounded Interval
Date
2002-01-02
Authors
Chmaj, Adam J. J.
Ren, Xiaofeng
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
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.
Description
Keywords
Local minimizers, Monotone solutions
Citation
Chmaj, A. J. J., & Ren, X. (2002). The nonlocal bistable equation: Stationary solutions on a bounded interval. <i>Electronic Journal of Differential Equations, 2002</i>(02), pp. 1-12.
Rights
Attribution 4.0 International