{"paper":{"title":"Sharp decay thresholds in weighted $L^\\infty$ for wave kinetic equations with power-law dispersion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Xilu Zhu","submitted_at":"2026-07-16T12:09:08Z","abstract_excerpt":"We study four-wave kinetic equations in space dimension three with power-law dispersion $\\omega(p)=|p|^a$ and collision kernels with high-frequency growth measured by $\\beta$. In weighted $L^\\infty$ spaces, we identify the sharp decay threshold $$ s_c=4\\beta+3-\\frac a2. $$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.14892","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.14892/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"}