{"paper":{"title":"The sharp estimate of the lifespan for the semilinear wave equation with time-dependent damping","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Masahiro Ikeda, Takahisa Inui","submitted_at":"2017-07-13T01:45:44Z","abstract_excerpt":"We consider the following semilinear wave equation with time-dependent damping. \\begin{align} \\tag{NLDW} \\left\\{ \\begin{array}{ll} \\partial_t^2 u - \\Delta u + b(t)\\partial_t u = |u|^{p}, & (t,x) \\in [0,T) \\times \\mathbb{R}^n, \\\\ u(0,x)=\\varepsilon u_0(x), u_t(0,x)=\\varepsilon u_1(x), & x \\in \\mathbb{R}^n, \\end{array} \\right. \\end{align} where $n \\in \\mathbb{N}$, $p>1$, $\\varepsilon>0$, and $b(t)\\thickapprox (t+1)^{-\\beta}$ with $\\beta \\in [-1,1)$. It is known that small data blow-up occurs when $1<p< p_F$ and, on the other hand, small data global existence holds when $p>p_F$, where $p_F:=1+2/n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03950","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"}