{"paper":{"title":"A Widder's type Theorem for the heat equation with nonlocal diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bego\\~na Barrios, Enrico Valdinoci, Fernando Soria, Ireneo Peral","submitted_at":"2013-02-07T15:54:05Z","abstract_excerpt":"The main goal of this work is to prove that every non-negative {\\it strong solution} $u(x,t)$ to the problem $$\n  u_t+(-\\Delta)^{\\alpha/2}u=0 \\ \\quad\\mbox{for }\n  (x,t)\\in\\mathbb{R}^{n}\\times(0,T), \\quad 0<\\alpha<2, $$\n  can be written as $$u(x,t)=\\int_{\\mathbb{R}^{n}}{P_{t}(x-y)u(y,0)\\, dy},$$ where $$P_{t}(x)=\\frac{1}{t^{n/\\alpha}}P\\left(\\frac{x}{t^{1/\\alpha}}\\right), $$ and $$ P(x):=\\int_{\\mathbb{R}^{n}}{e^{ix\\cdot\\xi-|\\xi|^{\\alpha}}d\\xi}. $$ This result shows uniqueness in the setting of non-negative solutions and extends some classical results for the heat equation by D. V. Widder in \\cit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.1786","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"}