{"paper":{"title":"Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.AP","authors_text":"Juraj F\\\"oldes, Peter Pol\\'a\\v{c}ik","submitted_at":"2013-11-27T17:49:51Z","abstract_excerpt":"We consider the Dirichlet problem  u_t &= \\Delta u + f(x, u, \\nabla u)+ h(x, t),& \\qquad &(x, t) \\in \\Omega \\times (0, \\infty), u &= 0, & \\qquad &(x, t) \\in \\partial\\Omega \\times (0, \\infty),  on a bounded domain $\\Omega \\subset \\mathbb{R}^N$. The domain and the nonlinearity $f$ are assumed to be invariant under the reflection about the $x_1$-axis, and the function $h$ accounts for a nonsymmetric decaying perturbation: $h(\\cdot, t)\\to 0$ as $t\\to\\infty$. In one of our main theorems, we prove the asymptotic symmetry of each bounded positive solution $u$. The novelty of this result is that the a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.7050","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"}