A counterexample to the Liouville property of some nonlocal problems

  • Julien Brasseur

    EHESS, CAMS, 54 Boulevard Raspail, F-75006 Paris, France
  • Jérôme Coville

    BioSP, INRAE, 84914, Avignon, France
A counterexample to the Liouville property of some nonlocal problems cover
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Abstract

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form

where is a bounded compact set, called an “obstacle”, and is a bistable nonlinearity. When is convex, it is known that solutions ranging in and satisfying as must be identically 1 in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data and for which this property fails.

Cite this article

Julien Brasseur, Jérôme Coville, A counterexample to the Liouville property of some nonlocal problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 3, pp. 549–579

DOI 10.1016/J.ANIHPC.2019.12.003