{"paper":{"title":"Transition semi-wave solutions of reaction diffusion equations with free boundaries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tao Zhou, Xing Liang","submitted_at":"2018-09-23T08:24:07Z","abstract_excerpt":"In this paper, we define the transition semi-wave solution of the following reaction diffusion equation with free boundaries \\begin{equation}\\label{0.1} \\left\\{\n  \\begin{aligned}\n  u_{t}=u_{xx}+f(t,x,u),\\ \\ &t\\in\\Real, x<h(t),\n  u(t,h(t))=0,\\ \\ &t\\in\\Real,\n  h^{\\prime}(t)=-\\mu u_{x}(t,h(t)),\\ \\ &t\\in\\Real,\n  \\end{aligned}\n  \\right. \\end{equation} In the homogeneous case, i.e., $f(t,x,u)=f(u)$, under the hypothesis $$ f(u)\\in {C}^{1}([0,1]), f(0)=f(1)=0, f^{\\prime}(1)<0, f(u)<0\\ \\text{for}\\ u>1, $$ we prove that the semi-wave connecting $1$ and $0$ is unique provided it exists. Furthermore, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.08551","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"}