{"paper":{"title":"$L^1$ estimates for oscillating integrals and their applications to semi-linear models with $\\sigma$-evolution like structural damping","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Michael Reissig, Tuan Anh Dao","submitted_at":"2018-08-15T11:54:35Z","abstract_excerpt":"The present paper is a continuation of our recent paper \\cite{DaoReissig}. We will consider the following Cauchy problems for semi-linear structurally damped $\\sigma$-evolution models: \\begin{equation*} u_{tt}+ (-\\Delta)^\\sigma u+ \\mu (-\\Delta)^\\delta u_t = f(u,u_t),\\, u(0,x)= u_0(x),\\, u_t(0,x)=u_1(x) \\end{equation*} with $\\sigma \\ge 1$, $\\mu>0$ and $\\delta \\in (\\frac{\\sigma}{2},\\sigma]$. Our aim is to study two main models including $\\sigma$-evolution models with structural damping $\\delta \\in (\\frac{\\sigma}{2},\\sigma)$ and those with visco-elastic damping $\\delta=\\sigma$. Here the function "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.05484","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"}