{"title":"具有Robin边界条件的反应扩散系统整体解的界","authors":"K. Kita, M. Otani","doi":"10.7153/dea-2019-11-09","DOIUrl":null,"url":null,"abstract":"In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we show that global solutions of this system are uniformly bounded in a suitable norm with respect to time. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted $L^1$ norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2018-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Bounds for global solutions of a reaction diffusion system with the Robin boundary conditions\",\"authors\":\"K. Kita, M. Otani\",\"doi\":\"10.7153/dea-2019-11-09\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we show that global solutions of this system are uniformly bounded in a suitable norm with respect to time. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted $L^1$ norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.\",\"PeriodicalId\":51863,\"journal\":{\"name\":\"Differential Equations & Applications\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2018-11-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Differential Equations & Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7153/dea-2019-11-09\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Equations & Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7153/dea-2019-11-09","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Bounds for global solutions of a reaction diffusion system with the Robin boundary conditions
In this paper, we consider the large-time behavior of solutions of a reaction diffusion system arising from a nuclear reactor model with the Robin boundary conditions, which consists of two real-valued unknown functions. In particular, we show that global solutions of this system are uniformly bounded in a suitable norm with respect to time. Since this system has no variational structure, we cannot apply the standard methods relying on the Lyapunov functional in order to obtain a priori estimates of global solutions. To cope with this difficulty, we make use of the weighted $L^1$ norm characterized by the first eigenfunction of Laplacian with the Robin boundary condition.