{"paper":{"title":"Some effects of the noise intensity upon non-linear stochastic heat equations on $[0,1]$","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Bin Xie","submitted_at":"2015-06-19T10:41:01Z","abstract_excerpt":"Various effects of the noise intensity upon the solution $u(t,x)$ of the stochastic heat equation with Dirichlet boundary conditions on $[0,1]$ are investigated. We show that for small noise intensity, the $p$-th moment of $\\sup_{x \\in [0,1]} |u(t,x)|$ is exponentially stable, however, for large one, it grows at least exponentially. We also prove that the noise excitation of the $p$-th energy of $u(t,x)$ is $4$, as the noise intensity goes to infinity. We formulate a common method to investigate the lower bounds of the above two different behaviors for large noise intensity, which are hard par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.05954","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"}