{"title":"基于半维里消失几何的Rd×T上聚焦能量临界非线性系统的长时间行为","authors":"Yongming Luo","doi":"10.1016/j.matpur.2023.07.006","DOIUrl":null,"url":null,"abstract":"<div><p><span>We study the focusing energy-critical NLS</span><span><span><span>(NLS)</span><span><math><mrow><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mi>u</mi><mo>=</mo><mo>−</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mi>u</mi></mrow></math></span></span></span> on the waveguide manifold <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span> with <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of <span>(NLS)</span> are purely determined by the semivirial-vanishing geometry which possesses an <em>energy-subcritical</em><span> characteristic. As a starting point, we consider a minimization problem </span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> defined on the semivirial-vanishing manifold with prescribed mass <em>c</em><span>. We prove that for all sufficiently large mass the variational problem </span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> has a unique optimizer <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> satisfying <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>, while for all sufficiently small mass, any optimizer of <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> must have non-trivial <em>y</em>-dependence. Afterwards, we prove that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> characterizes a sharp threshold for the bifurcation of finite time blow-up (<span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>) and globally scattering (<span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>) solutions of <span>(NLS)</span> in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"177 ","pages":"Pages 415-454"},"PeriodicalIF":2.1000,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On long time behavior of the focusing energy-critical NLS on Rd×T via semivirial-vanishing geometry\",\"authors\":\"Yongming Luo\",\"doi\":\"10.1016/j.matpur.2023.07.006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p><span>We study the focusing energy-critical NLS</span><span><span><span>(NLS)</span><span><math><mrow><mi>i</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mi>u</mi><mo>=</mo><mo>−</mo><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mfrac><mrow><mn>4</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup><mi>u</mi></mrow></math></span></span></span> on the waveguide manifold <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span> with <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of <span>(NLS)</span> are purely determined by the semivirial-vanishing geometry which possesses an <em>energy-subcritical</em><span> characteristic. As a starting point, we consider a minimization problem </span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> defined on the semivirial-vanishing manifold with prescribed mass <em>c</em><span>. We prove that for all sufficiently large mass the variational problem </span><span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> has a unique optimizer <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> satisfying <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>y</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><mn>0</mn></math></span>, while for all sufficiently small mass, any optimizer of <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> must have non-trivial <em>y</em>-dependence. Afterwards, we prove that <span><math><msub><mrow><mi>m</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> characterizes a sharp threshold for the bifurcation of finite time blow-up (<span><math><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn></math></span>) and globally scattering (<span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span>) solutions of <span>(NLS)</span> in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.</p></div>\",\"PeriodicalId\":51071,\"journal\":{\"name\":\"Journal de Mathematiques Pures et Appliquees\",\"volume\":\"177 \",\"pages\":\"Pages 415-454\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal de Mathematiques Pures et Appliquees\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782423001010\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001010","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On long time behavior of the focusing energy-critical NLS on Rd×T via semivirial-vanishing geometry
We study the focusing energy-critical NLS(NLS) on the waveguide manifold with . We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of (NLS) are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem defined on the semivirial-vanishing manifold with prescribed mass c. We prove that for all sufficiently large mass the variational problem has a unique optimizer satisfying , while for all sufficiently small mass, any optimizer of must have non-trivial y-dependence. Afterwards, we prove that characterizes a sharp threshold for the bifurcation of finite time blow-up () and globally scattering () solutions of (NLS) in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.