{"paper":{"title":"Remarks on contractions of reaction-diffusion PDE's on weighted L^2 norms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"cs.SY","authors_text":"Zahra Aminzare","submitted_at":"2012-08-05T19:58:37Z","abstract_excerpt":"In [1], we showed contractivity of reaction-diffusion PDE: \\frac{\\partial u}{\\partial t}({\\omega},t) = F(u({\\omega},t)) + D\\Delta u({\\omega},t) with Neumann boundary condition, provided \\mu_{p,Q}(J_F (u)) < 0 (uniformly on u), for some 1 \\leq p \\leq \\infty and some positive, diagonal matrix Q, where J_F is the Jacobian matrix of F. This note extends the result for Q weighted L_2 norms, where Q is a positive, symmetric (not merely diagonal) matrix and Q^2D+DQ^2>0."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.1045","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"}