{"paper":{"title":"Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Johannes Lankeit","submitted_at":"2015-11-05T20:37:25Z","abstract_excerpt":"We prove convergence of positive solutions to \\[ u_t = u\\Delta u + u\\int_{\\Omega} |\\nabla u|^2, \\qquad u\\rvert_{\\partial\\Omega} =0, \\qquad u(\\cdot,0)=u_0 \\] in a bounded domain $\\Omega\\subset \\mathbb{R}^n$, $n\\ge 1$, with smooth boundary in the case of $\\int_\\Omega u_0=1$ and identify the $W_0^{1,2}(\\Omega)$-limit of $u(t)$ as $t\\to \\infty$ as the solution of the corresponding stationary problem. This behaviour is different from the cases of $\\int_\\Omega u_0<1$ and $\\int_\\Omega u_0>1$ which are known to result in convergence to zero or blow-up in finite time, respectively.\n\n  The proof is base"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01885","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"}