{"paper":{"title":"On continuation properties after blow-up time for $L^2$-critical gKdV equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Yang Lan","submitted_at":"2017-09-25T19:37:13Z","abstract_excerpt":"In this paper, we consider a blow-up solution $u(t)$ to the $L^2$-critical gKdV equation $\\partial_tu+(u_{xx}+u^5)_x=0$, with finite blow-up time $T<+\\infty$. We expect to construct a natural extension of $u(t)$ after the blow-up time. To do this, we consider the solution $u_{\\gamma}(t)$ to the saturated $L^2$-critical gKdV equation $\\partial_tu+(u_{xx}+u^5-\\gamma u|u|^{q-1})_x=0$ with the same initial data, where $\\gamma>0$ and $q>5$. A standard argument shows that $u_{\\gamma}(t)$ is always global in time and for all $t<T$, $u_{\\gamma}(t)$ converges to $u(t)$ in $H^1$ as $\\gamma\\rightarrow0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.09535","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"}