{"paper":{"title":"Backward blow-up estimates and initial trace for a parabolic system of reaction-diffusion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cecilia Yarur (Departamento de Matematicas y CC), Marie-Fran\\c{c}oise Bidaut-V\\'eron (LMPT), Marta Garcia-Huidobro","submitted_at":"2010-03-22T15:16:21Z","abstract_excerpt":"In this article we study the positive solutions of the parabolic semilinear system of competitive type \\[ \\left\\{\\begin{array} [c]{c}% u_{t}-\\Delta u+v^{p}=0, v_{t}-\\Delta v+u^{q}=0, \\end{array} \\right. \\] in $\\Omega\\times\\left(0,T\\right) $, where $\\Omega$ is a domain of $\\mathbb{R}^{N},$ and $p,q>0,$ $pq\\neq1.$ Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case $pq>1$ of the form \\[ u(x,t)\\leqq Ct^{-(p+1)/(pq-1)},\\qquad v(x,t)\\leqq Ct^{-(q+1)/(pq-1)}% \\] in $\\omega\\times\\left(0,T_{1}\\right) ,$ for any domain $\\omega \\subset\\subset\\Omega$ and $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1003.4189","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"}