{"title":"Decay estimates for Beam equations with potential in dimension three","authors":"Miao Chen , Ping Li , Avy Soffer , Xiaohua Yao","doi":"10.1016/j.jfa.2024.110671","DOIUrl":null,"url":null,"abstract":"<div><p>This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><mo>(</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi><mo>)</mo><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace><mi>u</mi><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span></span></span> in dimension three, where <em>V</em> is a real-valued and decaying potential. Assume that zero is a regular point of <span><math><mi>H</mi><mo>=</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>V</mi></math></span>, we first prove the following optimal time decay estimates of the solution operators<span><span><span><math><msub><mrow><mo>‖</mo><mi>cos</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≲</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mspace></mspace><mspace></mspace><mtext>and</mtext><msub><mrow><mo>‖</mo><mfrac><mrow><mi>sin</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac><msub><mrow><mi>P</mi></mrow><mrow><mi>a</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>→</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≲</mo><mo>|</mo><mi>t</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>.</mo></math></span></span></span> Moreover, if zero is a resonance of <em>H</em>, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators <span><math><mi>cos</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></math></span> and <span><math><mfrac><mrow><mi>sin</mi><mo></mo><mo>(</mo><mi>t</mi><msqrt><mrow><mi>H</mi></mrow></msqrt><mo>)</mo></mrow><mrow><msqrt><mrow><mi>H</mi></mrow></msqrt></mrow></mfrac></math></span>, but their decay will be significantly changed for the second and third-kind resonances.</p></div>","PeriodicalId":15750,"journal":{"name":"Journal of Functional Analysis","volume":null,"pages":null},"PeriodicalIF":1.7000,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022123624003598","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is devoted to studying time decay estimates of the solution for Beam equation (higher order type wave equation) with a potential in dimension three, where V is a real-valued and decaying potential. Assume that zero is a regular point of , we first prove the following optimal time decay estimates of the solution operators Moreover, if zero is a resonance of H, then time decay of the solution operators also is considered. It is noted that a first-kind resonance does not affect the decay rates of the propagator operators and , but their decay will be significantly changed for the second and third-kind resonances.
期刊介绍:
The Journal of Functional Analysis presents original research papers in all scientific disciplines in which modern functional analysis plays a basic role. Articles by scientists in a variety of interdisciplinary areas are published.
Research Areas Include:
• Significant applications of functional analysis, including those to other areas of mathematics
• New developments in functional analysis
• Contributions to important problems in and challenges to functional analysis