{"title":"Local-in-space wave breaking criteria for a generalized rod equation","authors":"","doi":"10.1016/j.jde.2024.08.035","DOIUrl":null,"url":null,"abstract":"<div><p>A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line <span><math><mi>R</mi></math></span>. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>)</mo></math></span> at a single local point <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>R</mi></math></span>.</p></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022039624005175","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
A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line . Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values and at a single local point .
期刊介绍：
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics
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