{"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}
引用次数: 0
Abstract
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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