{"paper":{"title":"Sharp decay rate for the damped wave equation with convex-shaped damping","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.AP","authors_text":"Chenmin Sun","submitted_at":"2021-06-22T13:47:25Z","abstract_excerpt":"We revisit the damped wave equation on two-dimensional torus where the damped region does not satisfy the geometric control condition. We show that if the damping vanishes as a H\\\"older function $|x|^{\\beta}$, and in addition, the boundary of the damped region is strictly convex, the wave is stable at rate $t^{-1+\\frac{2}{2\\beta+7}}$, which is better than the known optimal decay rate $t^{-1+\\frac{1}{\\beta+3}}$ for strip-shaped dampings of the same H\\\"older regularity. Moreover, we show by example that the decay rate is optimal. This illustrates the fact that the energy decay rate depends not o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2106.11782","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2106.11782/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}