{"title":"贝索夫空间中福恩贝格-惠瑟姆型方程的良好拟合性和对初始数据的非均匀依赖性","authors":"Xueyuan Qi","doi":"10.1007/s00605-024-01974-y","DOIUrl":null,"url":null,"abstract":"<p>In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces <span>\\(B^{s}_{p,r}({\\mathbb {R}})\\)</span> with <span>\\( 1\\le p,r\\le \\infty \\)</span> and <span>\\(s> max\\{1+\\frac{1}{p},\\frac{3}{2}\\}\\)</span>, which improve the previous work in Sobolev spaces <span>\\( H^{s}({\\mathbb {R}})= B^{s}_{2,2}({\\mathbb {R}})\\)</span> with <span>\\( s>\\frac{3}{2}\\)</span> (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces <span>\\(B^{s}_{p,r}({\\mathbb {R}})\\)</span> with <span>\\( 1\\le p\\le \\infty \\)</span>,<span>\\( 1\\le r< \\infty \\)</span> and <span>\\(s> max\\{1+\\frac{1}{p},\\frac{3}{2}\\}\\)</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"10 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces\",\"authors\":\"Xueyuan Qi\",\"doi\":\"10.1007/s00605-024-01974-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces <span>\\\\(B^{s}_{p,r}({\\\\mathbb {R}})\\\\)</span> with <span>\\\\( 1\\\\le p,r\\\\le \\\\infty \\\\)</span> and <span>\\\\(s> max\\\\{1+\\\\frac{1}{p},\\\\frac{3}{2}\\\\}\\\\)</span>, which improve the previous work in Sobolev spaces <span>\\\\( H^{s}({\\\\mathbb {R}})= B^{s}_{2,2}({\\\\mathbb {R}})\\\\)</span> with <span>\\\\( s>\\\\frac{3}{2}\\\\)</span> (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces <span>\\\\(B^{s}_{p,r}({\\\\mathbb {R}})\\\\)</span> with <span>\\\\( 1\\\\le p\\\\le \\\\infty \\\\)</span>,<span>\\\\( 1\\\\le r< \\\\infty \\\\)</span> and <span>\\\\(s> max\\\\{1+\\\\frac{1}{p},\\\\frac{3}{2}\\\\}\\\\)</span>.</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":\"10 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-04-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-01974-y\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01974-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
本文首先在贝索夫空间 \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p,r\le \infty \) and\(s>;max{1+\frac{1}{p},\frac{3}{2}\}), which improve the previous work in Sobolev spaces \( H^{s}({\mathbb {R}})= B^{s}_{2,2}({\mathbb {R}})\) with\( s>\frac{3}{2}\) (Lai and Luo in J Differ Equ 344:509-521, 2023).此外,我们还证明了在贝索夫空间 \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p\le \infty \),\( 1\le r< \infty \) and\(s> max\{1+\frac{1}{p},\frac{3}{2}\}) 中,解并非均匀连续地依赖于初始数据。
Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces
In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p,r\le \infty \) and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\), which improve the previous work in Sobolev spaces \( H^{s}({\mathbb {R}})= B^{s}_{2,2}({\mathbb {R}})\) with \( s>\frac{3}{2}\) (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) with \( 1\le p\le \infty \),\( 1\le r< \infty \) and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\).