{"paper":{"title":"Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kyeong-Hun Kim, Sungbin Lim","submitted_at":"2015-04-28T09:13:35Z","abstract_excerpt":"Let $p(t,x)$ be the fundamental solution to the problem $$ \\partial_{t}^{\\alpha}u=-(-\\Delta)^{\\beta}u, \\quad \\alpha\\in (0,2), \\, \\beta\\in (0,\\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-\\Delta_x)^{\\gamma}D_{t}^{\\sigma}I_{t}^{\\delta}p(t,x), \\quad \\forall\\,\\, n\\in\\mathbb{Z}_{+}, \\,\\, \\gamma\\in[0,\\beta],\\,\\, \\sigma, \\delta \\in[0,\\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-\\Delta_x)^{\\gamma}$ is a fractional Laplace operator and $D_{t}^{\\sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07386","kind":"arxiv","version":4},"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"}