{"paper":{"title":"Slowly oscillating wavefronts of the KPP-Fisher delayed equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Karel Hasik, Sergei Trofimchuk","submitted_at":"2012-06-03T20:18:12Z","abstract_excerpt":"This paper concerns the semi-wavefronts (i.e. bounded solutions $u=\\phi(x \\nu +ct) >0,$ $ |\\nu|=1, $ satisfying $\\phi(-\\infty)=0$) to the delayed KPP-Fisher equation $$u_t(t,x) = \\Delta u(t,x) + u(t,x)(1-u(t-\\tau,x)), \\ u \\geq 0,\\ x \\in \\R^m. \\eqno(*)$$ First, we show that each semi-wavefront should be either monotone or slowly oscillating. Then a complete solution to the problem of existence of semi-wavefronts is provided. We prove next that the semi-wavefronts are in fact wavefronts (i.e. additionally $\\phi(+\\infty)=1$) if $c \\geq 2$ and $\\tau \\leq 1$; our proof uses dynamical properties of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.0484","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"}