{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:YZNZFZDIO6ZM5CQJ2S4DAZJSTT","short_pith_number":"pith:YZNZFZDI","schema_version":"1.0","canonical_sha256":"c65b92e46877b2ce8a09d4b83065329cce30f16df8efa83c7e12ce98c1d7cb4d","source":{"kind":"arxiv","id":"1809.08551","version":1},"attestation_state":"computed","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. 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