{"paper":{"title":"Asymptotic behavior and representation of solutions to a Volterra kind of equation with a singular kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mahamadi Warma, Rodrigo Ponce","submitted_at":"2016-10-27T12:39:09Z","abstract_excerpt":"Let $A$ be a densely defined closed, linear $\\omega$-sectorial operator of angle $\\theta\\in [0,\\frac{\\pi}{2})$ on a Banach space $X$ for some $\\omega\\in\\mathbb R$. We give an explicit representation (in terms of some special functions) and study the precise asymptotic behavior as time goes to infinity of solutions to the following diffusion equation with memory: $\\displaystyle u'(t)=Au(t)+(\\kappa\\ast Au)(t), \\, t >0$, $u(0)=u_0$, associated with the (possible) singular kernel\n  $\\kappa(t)=\\alpha e^{-\\beta t}\\frac{t^{\\mu-1}}{\\Gamma(\\mu)},\\;\\;t>0$, where $\\alpha\\in\\R$, $\\alpha\\ne 0$, $\\beta\\ge 0"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.08750","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"}