{"title":"一个","authors":"Hiroki Yagisita","doi":"10.1525/9780520965553-003","DOIUrl":null,"url":null,"abstract":"We consider the nonlocal analogue of the Fisher-KPP equation where $ \\mu$ is a Borel-measure on $ \\mathbb{R}$ with $ \\mu(\\mathbb{R})=1$ and $f$ satisfies $f(0)=f(1)=$ $0$ and $f >0$ in $(0,1)$ . We do not assume that $ \\mu$ is absolutely continuous. The equation may have a standing wave solution (a traveling wave solution with speed $0$ ) whose profile is a monotone but discontinuous function. We show that there is a constant $c _{*}$ such that it has a traveling wave solution with monotone profile and speed $c$ when $c \\geq c_{*}$ while no periodic traveling wave solution with average speed $c$ when $c <c_{*}$ . In order to prove it, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semfflow generated by the equation does not have compactness with respect to the compact-open topology.","PeriodicalId":149341,"journal":{"name":"Dictionary of the Ben cao gang mu, Volume 2","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A\",\"authors\":\"Hiroki Yagisita\",\"doi\":\"10.1525/9780520965553-003\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the nonlocal analogue of the Fisher-KPP equation where $ \\\\mu$ is a Borel-measure on $ \\\\mathbb{R}$ with $ \\\\mu(\\\\mathbb{R})=1$ and $f$ satisfies $f(0)=f(1)=$ $0$ and $f >0$ in $(0,1)$ . We do not assume that $ \\\\mu$ is absolutely continuous. The equation may have a standing wave solution (a traveling wave solution with speed $0$ ) whose profile is a monotone but discontinuous function. We show that there is a constant $c _{*}$ such that it has a traveling wave solution with monotone profile and speed $c$ when $c \\\\geq c_{*}$ while no periodic traveling wave solution with average speed $c$ when $c <c_{*}$ . In order to prove it, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semfflow generated by the equation does not have compactness with respect to the compact-open topology.\",\"PeriodicalId\":149341,\"journal\":{\"name\":\"Dictionary of the Ben cao gang mu, Volume 2\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-12-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Dictionary of the Ben cao gang mu, Volume 2\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1525/9780520965553-003\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dictionary of the Ben cao gang mu, Volume 2","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1525/9780520965553-003","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We consider the nonlocal analogue of the Fisher-KPP equation where $ \mu$ is a Borel-measure on $ \mathbb{R}$ with $ \mu(\mathbb{R})=1$ and $f$ satisfies $f(0)=f(1)=$ $0$ and $f >0$ in $(0,1)$ . We do not assume that $ \mu$ is absolutely continuous. The equation may have a standing wave solution (a traveling wave solution with speed $0$ ) whose profile is a monotone but discontinuous function. We show that there is a constant $c _{*}$ such that it has a traveling wave solution with monotone profile and speed $c$ when $c \geq c_{*}$ while no periodic traveling wave solution with average speed $c$ when $c
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