带有高导数项的波方程的 L2$L^{2}$ 生长特性

IF 0.8 3区 数学 Q2 MATHEMATICS
Xiaoyan Li, Ryo Ikehata
{"title":"带有高导数项的波方程的 L2$L^{2}$ 生长特性","authors":"Xiaoyan Li,&nbsp;Ryo Ikehata","doi":"10.1002/mana.202300358","DOIUrl":null,"url":null,"abstract":"<p>We consider the Cauchy problem in <span></span><math>\n <semantics>\n <msup>\n <mi>R</mi>\n <mi>n</mi>\n </msup>\n <annotation>${\\bf R}^{n}$</annotation>\n </semantics></math> for the wave equation with a higher derivative term. We derive sharp growth estimates of the <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mn>2</mn>\n </msup>\n <annotation>$L^{2}$</annotation>\n </semantics></math>-norm of the solution itself for the case of <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$n = 1$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n = 2$</annotation>\n </semantics></math>. By imposing the weighted <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mn>1</mn>\n </msup>\n <annotation>$L^{1}$</annotation>\n </semantics></math>-initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$n\\ge 3$</annotation>\n </semantics></math>, we observe that the <span></span><math>\n <semantics>\n <msup>\n <mi>L</mi>\n <mn>2</mn>\n </msup>\n <annotation>$L^{2}$</annotation>\n </semantics></math>-growth behavior of the solution never occurs in the <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msup>\n <mi>L</mi>\n <mn>2</mn>\n </msup>\n <mo>∩</mo>\n <msup>\n <mi>L</mi>\n <mn>1</mn>\n </msup>\n <mo>)</mo>\n </mrow>\n <annotation>$(L^{2}\\cap L^{1})$</annotation>\n </semantics></math>-framework of the initial data.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"297 10","pages":"3625-3640"},"PeriodicalIF":0.8000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"L\\n 2\\n \\n $L^{2}$\\n -growth property for the wave equation with a higher derivative term\",\"authors\":\"Xiaoyan Li,&nbsp;Ryo Ikehata\",\"doi\":\"10.1002/mana.202300358\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider the Cauchy problem in <span></span><math>\\n <semantics>\\n <msup>\\n <mi>R</mi>\\n <mi>n</mi>\\n </msup>\\n <annotation>${\\\\bf R}^{n}$</annotation>\\n </semantics></math> for the wave equation with a higher derivative term. We derive sharp growth estimates of the <span></span><math>\\n <semantics>\\n <msup>\\n <mi>L</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$L^{2}$</annotation>\\n </semantics></math>-norm of the solution itself for the case of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n <annotation>$n = 1$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>=</mo>\\n <mn>2</mn>\\n </mrow>\\n <annotation>$n = 2$</annotation>\\n </semantics></math>. By imposing the weighted <span></span><math>\\n <semantics>\\n <msup>\\n <mi>L</mi>\\n <mn>1</mn>\\n </msup>\\n <annotation>$L^{1}$</annotation>\\n </semantics></math>-initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>≥</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$n\\\\ge 3$</annotation>\\n </semantics></math>, we observe that the <span></span><math>\\n <semantics>\\n <msup>\\n <mi>L</mi>\\n <mn>2</mn>\\n </msup>\\n <annotation>$L^{2}$</annotation>\\n </semantics></math>-growth behavior of the solution never occurs in the <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msup>\\n <mi>L</mi>\\n <mn>2</mn>\\n </msup>\\n <mo>∩</mo>\\n <msup>\\n <mi>L</mi>\\n <mn>1</mn>\\n </msup>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(L^{2}\\\\cap L^{1})$</annotation>\\n </semantics></math>-framework of the initial data.</p>\",\"PeriodicalId\":49853,\"journal\":{\"name\":\"Mathematische Nachrichten\",\"volume\":\"297 10\",\"pages\":\"3625-3640\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematische Nachrichten\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300358\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.202300358","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

我们考虑了带有高导数项的波方程中的考希问题。我们推导出在 和 的情况下,解本身的-正值的急剧增长估计值。通过施加加权初速度,我们可以得到解本身的下限和上限估计值。对于 和 的情况,我们发现解的增长行为从未出现在初始数据的框架中。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
L 2 $L^{2}$ -growth property for the wave equation with a higher derivative term

We consider the Cauchy problem in R n ${\bf R}^{n}$ for the wave equation with a higher derivative term. We derive sharp growth estimates of the L 2 $L^{2}$ -norm of the solution itself for the case of n = 1 $n = 1$ and n = 2 $n = 2$ . By imposing the weighted L 1 $L^{1}$ -initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of n 3 $n\ge 3$ , we observe that the L 2 $L^{2}$ -growth behavior of the solution never occurs in the ( L 2 L 1 ) $(L^{2}\cap L^{1})$ -framework of the initial data.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.50
自引率
0.00%
发文量
157
审稿时长
4-8 weeks
期刊介绍: Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信