{"paper":{"title":"Representation of solutions and large-time behavior for fully nonlocal diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juhana Siljander, Jukka Kemppainen, Rico Zacher","submitted_at":"2015-05-11T20:56:21Z","abstract_excerpt":"We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems:\n  (i) a representation formula for classical solutions,\n  (ii) a quantitative decay rate at which the solution tends to the fundamental solution,\n  (iii) optimal $L^2$-decay of mild solutions in all dimensions,\n  (iv) $L^2$-decay of weak solutions via energy methods.\n  The first result relies on a delicate analysis of the definition of classical solutions. After proving the representation formula we carefully analyze the integral representation to obtain t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02803","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"}