{"paper":{"title":"A counterexample to the Liouville property of some nonlocal problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.AP","authors_text":"INRA), J\\'er\\^ome Coville (BIOSP), Julien Brasseur (I2M","submitted_at":"2018-04-20T08:30:44Z","abstract_excerpt":"In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$$ \\int\\_{\\mathbb{R}^N\\setminus K} J(x-y)\\,( u(y)-u(x) )\\mathrm{d}y+f(u(x))=0, \\quad x\\in\\R^N\\setminus K,$$where $K\\subset\\mathbb{R}^N$ is a bounded compact set, called an \"obstacle\", and $f$ is a bistable nonlinearity. When $K$ is convex, it is known that solutions ranging in $[0,1]$ and satisfying $u(x)\\to1$ as $|x|\\to\\infty$ must be identically $1$ in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data $f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07485","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}