{"paper":{"title":"Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bendong Lou, Maolin Zhou, Thomas Giletti, Xinfu Chen","submitted_at":"2013-09-28T08:44:04Z","abstract_excerpt":"We study the long time behavior, as $t\\to\\infty$, of solutions of $$ \\left\\{ \\begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \\ t >0,\\\\ u(0,t) = b u_x(0,t), & t>0,\\\\ u(x,0) = u_0 (x)\\geqslant 0 , & x\\geqslant 0, \\end{array} \\right. $$ where $b\\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\\{ \\sigma \\phi \\}_{\\sigma >0}$, where $\\phi$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $\\sigma^*$ suc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.7441","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"}