{"paper":{"title":"Structure of $\\omega$-limit Sets for Almost-periodic Parabolic Equations on $S^1$ with Reflection Symmetry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Dun Zhou, Wenxian Shen, Yi Wang","submitted_at":"2016-01-19T13:07:49Z","abstract_excerpt":"The structure of the $\\omega$-limit sets is thoroughly investigated for the skew-product semiflow which is generated by a scalar reaction-diffusion equation \\begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\\,\\,t>0,\\,x\\in S^{1}=\\mathbb{R}/2\\pi \\mathbb{Z}, \\end{equation*} where $f$ is uniformly almost periodic in $t$ and satisfies $f(t,u,u_x)=f(t,u,-u_x)$. We show that any $\\omega$-limit set $\\Omega$ contains at most two minimal sets. Moreover, any hyperbolic $\\omega$-limit set $\\Omega$ is a spatially-homogeneous $1$-cover of hull $H(f)$. When $\\dim V^c(\\Omega)=1$ ($V^c(\\Omega)$ is the center space a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.04906","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"}