{"paper":{"title":"Two-end solutions to the Allen-Cahn equation in $\\mathbb{R}^{3}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.AP","authors_text":"Changfeng Gui, Juncheng Wei, Yong Liu","submitted_at":"2015-02-20T18:21:26Z","abstract_excerpt":"In this paper we consider the Allen-Cahn equation $$ -\\Delta u = u-u^3 \\ \\mbox{in} \\ {\\mathbb R}^3 $$ We prove that for each $k\\in\\left( \\sqrt{2},+\\infty\\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \\| u-H(\\cdot -k \\ln r + c_k) \\|_{L^\\infty} \\to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.05963","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"}