{"paper":{"title":"Scattering for the beam equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Yifei Wu","submitted_at":"2011-08-03T11:42:15Z","abstract_excerpt":"In this paper, we study the scattering for the nonlinear beam\nequation $u_{tt}+\\Delta^2u+mu+\\mu |u|^{p-1}u=0$. Our results include\ntwo aspects. In the defocusing case we show that the scattering\nholds for $d=1$, which extends the result in \\cite{Pau-Beam} to one\ndimension. In the focusing case, we show that the scattering holds\nin $\\R^d (d\\ge 1)$ when the energy $E(u_0,u_1)<E(R,0)$ and $\\|\\Delta\nu_0\\|_{L^2}^2+m\\|u_0\\|_{L^2}^2<\\|\\Delta R\\|_{L^2}^2+m\\|R\\|_{L^2}^2$\nfor ground state $R$. The difficulties lie the absence of the\nscaling invariance and a Galilean transformation for the equation to\nco"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.0825","kind":"arxiv","version":2},"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"}