{"title":"Evolution of the radius of analyticity for the generalized Benjamin equation","authors":"Renata O. Figueira, M. Panthee","doi":"10.3934/dcds.2023039","DOIUrl":null,"url":null,"abstract":"In this work we consider the initial value problem for the generalized Benjamin equation \\begin{equation}\\label{Benj-IVP} \\begin{cases} \\partial_t u-l\\mathcal{H} \\partial_x^2u-\\partial_x^3u+u^p\\partial_xu = 0, \\quad x,\\; t\\in \\mathbb{R};\\;\\;,\\; p\\geq 1, \\\\ u(x,0) = u_0(x), \\end{cases} \\end{equation} where $u=u(x,t)$ is a real valued function, $0<l<1$ and $\\mathcal{H}$ is the Hilbert transform. This model was introduced by T. B. Benjamin (J. Fluid Mech. 245 (1992) 401--411) and describes unidirectional propagation of long waves in a two-fluid system where the lower fluid with greater density is infinitely deep and the interface is subject to capillarity. We prove that the local solution to the IVP associated with the generalized Benjamin equation for given data in the spaces of functions analytic on a strip around the real axis continue to be analytic without shrinking the width of the strip in time. We also study the evolution in time of the radius of spatial analyticity and show that it can decrease as the time advances. Finally, we present an algebraic lower bound on the possible rate of decrease in time of the uniform radius of spatial analyticity.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete and Continuous Dynamical Systems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/dcds.2023039","RegionNum":3,"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
In this work we consider the initial value problem for the generalized Benjamin equation \begin{equation}\label{Benj-IVP} \begin{cases} \partial_t u-l\mathcal{H} \partial_x^2u-\partial_x^3u+u^p\partial_xu = 0, \quad x,\; t\in \mathbb{R};\;\;,\; p\geq 1, \\ u(x,0) = u_0(x), \end{cases} \end{equation} where $u=u(x,t)$ is a real valued function, $0
期刊介绍:
DCDS, series A includes peer-reviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers.