{"paper":{"title":"Pointwise Bounds and Blow-up for Nonlinear Fractional Parabolic Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Steven D. Taliaferro","submitted_at":"2019-01-28T19:44:54Z","abstract_excerpt":"We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \\begin{equation}\\label{0.1}\n  0\\leq(\\partial_t-\\Delta)^\\alpha u\\leq u^\\lambda \\quad\\text{ in }\\mathbb{R}^n \\times\\mathbb{R},\\,n\\geq1, \\end{equation} \\begin{equation}\\label{0.2}\n  u=0\\quad\\text{in }\\mathbb{R}^n\\times(-\\infty,0) \\end{equation} where $\\lambda$ and $\\alpha$ are positive constants. To do this we first give a definition---tailored for our study of this problem---of fractional powers of the heat operator $(\\partial_t-\\Delta)^\\alpha :Y\\to X$ where $X$ and $Y$ are linear spac"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.09964","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"}