{"title":"带有高导数项的波方程的 L2$L^{2}$ 生长特性","authors":"Xiaoyan Li, 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, 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}
L
2
$L^{2}$
-growth property for the wave equation with a higher derivative term
We consider the Cauchy problem in for the wave equation with a higher derivative term. We derive sharp growth estimates of the -norm of the solution itself for the case of and . By imposing the weighted -initial velocity, we can get the lower and upper bound estimates of the solution itself. For the case of , we observe that the -growth behavior of the solution never occurs in the -framework of the initial data.
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