{"paper":{"title":"Existence and nonuniqueness of segregated solutions to a class of cross-diffusion systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gonzalo Galiano, Juli\\'an Velasco, Sergey Shmarev","submitted_at":"2013-11-14T10:57:11Z","abstract_excerpt":"We study the the Dirichlet problem for the cross-diffusion system \\[ \\partial_tu_i=\\operatorname{div}\\left(a_iu_i\\nabla (u_1+u_2)\\right)+f_i(u_1,u_2),\\quad i=1,2,\\quad a_i=const>0, \\] in the cylinder $Q=\\Omega\\times (0,T]$. The functions $f_i$ are assumed to satisfy the conditions $f_1(0,r)=0$, $f_2(s,0)=0$, $f_1(0,r)$, $f_2(s,0)$ are locally Lipschitz-continuous. It is proved that for suitable initial data $u_0$, $v_0$ the system admits segregated solutions $(u_1,u_2)$ such that $u_i\\in L^{\\infty}(Q)$, $u_1+u_2\\in C^{0}(\\overline{Q})$, $u_1+u_2>0$ and $u_1\\cdot u_2=0$ everywhere in $Q$. We sh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.3454","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"}