{"title":"论圆柱体上二维立方非线性薛定谔方程的全局和奇异动力学","authors":"Adán J. Corcho , Mahendra Panthee","doi":"10.1016/j.na.2024.113519","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the Cauchy problem associated to the focusing cubic nonlinear Schrödinger equation posed on a two dimensional cylindrical domain <span><math><mrow><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math></span>. We prove that localized transverse perturbations of an especial one-parameter family of bound states solutions <span><math><mrow><mo>{</mo><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>ω</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></mrow><mo>}</mo></mrow></math></span>, <span><math><mrow><mi>ω</mi><mo>></mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span> can be extended globally in time. On the other hand, we establish the existence of solution in the energy space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, with non-critical mass, that blows-up in finite time under the hypothesis of no growth in time of the directional <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-norm of the solution when the periodic variable <span><math><mi>y</mi></math></span> is localized. We also prove that a family of bound states <span><math><mrow><mo>{</mo><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>ω</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></mrow><mo>}</mo></mrow></math></span> is not uniformly continuous from <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into the space of continuous functions <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>;</mo><mspace></mspace><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span>, whenever <span><math><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>≤</mo><mi>s</mi><mo><</mo><mn>0</mn></mrow></math></span>, including the regularity <span><math><mrow><mi>s</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> for the <em>non-uniformly continuous flow</em>, unlike to the case of focusing cubic nonlinear Schrödinger equation on <span><math><mi>R</mi></math></span>.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the global and singular dynamics of the 2D cubic nonlinear Schrödinger equation on cylinders\",\"authors\":\"Adán J. Corcho , Mahendra Panthee\",\"doi\":\"10.1016/j.na.2024.113519\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We consider the Cauchy problem associated to the focusing cubic nonlinear Schrödinger equation posed on a two dimensional cylindrical domain <span><math><mrow><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math></span>. We prove that localized transverse perturbations of an especial one-parameter family of bound states solutions <span><math><mrow><mo>{</mo><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>ω</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></mrow><mo>}</mo></mrow></math></span>, <span><math><mrow><mi>ω</mi><mo>></mo><mo>−</mo><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mrow></math></span> can be extended globally in time. On the other hand, we establish the existence of solution in the energy space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>, with non-critical mass, that blows-up in finite time under the hypothesis of no growth in time of the directional <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-norm of the solution when the periodic variable <span><math><mi>y</mi></math></span> is localized. We also prove that a family of bound states <span><math><mrow><mo>{</mo><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>ω</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></mrow><mo>}</mo></mrow></math></span> is not uniformly continuous from <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> into the space of continuous functions <span><math><mrow><mi>C</mi><mrow><mo>(</mo><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>;</mo><mspace></mspace><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mrow><mo>(</mo><mi>R</mi><mo>×</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span>, whenever <span><math><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>≤</mo><mi>s</mi><mo><</mo><mn>0</mn></mrow></math></span>, including the regularity <span><math><mrow><mi>s</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></math></span> for the <em>non-uniformly continuous flow</em>, unlike to the case of focusing cubic nonlinear Schrödinger equation on <span><math><mi>R</mi></math></span>.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-02-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24000385\",\"RegionNum\":2,\"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":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24000385","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
摘要
我们考虑了在二维圆柱域 R×Tℓ 上提出的与聚焦立方非线性薛定谔方程相关的柯西问题。我们证明了边界态解 {uω,ℓ}, ω>-π2ℓ2 的特殊单参数族的局部横向扰动可以在时间上进行全局扩展。另一方面,我们在能量空间 H1(R×Tℓ) 中建立了具有非临界质量的解的存在性,当周期变量 y 局部化时,在解的方向 Lx2-norm 不随时间增长的假设条件下,该解在有限时间内炸毁。我们还证明了当-1/2≤s<0时,从 Hs(R×Tℓ)到连续函数空间 C([0,T];Hs(R×Tℓ)) 的束缚态{uω,ℓ}族不是均匀连续的,包括非均匀连续流的正则性 s=-12,这与 R 上聚焦立方非线性薛定谔方程的情况不同。
On the global and singular dynamics of the 2D cubic nonlinear Schrödinger equation on cylinders
We consider the Cauchy problem associated to the focusing cubic nonlinear Schrödinger equation posed on a two dimensional cylindrical domain . We prove that localized transverse perturbations of an especial one-parameter family of bound states solutions , can be extended globally in time. On the other hand, we establish the existence of solution in the energy space , with non-critical mass, that blows-up in finite time under the hypothesis of no growth in time of the directional -norm of the solution when the periodic variable is localized. We also prove that a family of bound states is not uniformly continuous from into the space of continuous functions , whenever , including the regularity for the non-uniformly continuous flow, unlike to the case of focusing cubic nonlinear Schrödinger equation on .
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