{"title":"Some new asymptotic behaviors of a two-component b-family equations","authors":"Lijun Du, Xinglong Wu","doi":"10.1007/s10231-024-01429-z","DOIUrl":null,"url":null,"abstract":"<div><p>In this article, we investigate some new asymptotic behaviors of the solution for a two-component <i>b</i>-family equations. We first prove the persistence properties of the solution for Eq. (1.1) when the initial data decay logarithmically, algebraically at infinity with the power <span>\\(\\beta \\in (0,\\infty )\\)</span>. Subsequently, we obtain the infinite propagation of the solution to Eq. (1.1). If the initial data satisfy certain compact condition, then the nontrivial solution <i>u</i> of Eq. (1.1) immediately loses compactly supported. Meanwhile, the solution <i>u</i> decays exponentially as <span>\\(|x|\\rightarrow \\infty \\)</span>.</p></div>","PeriodicalId":8265,"journal":{"name":"Annali di Matematica Pura ed Applicata","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annali di Matematica Pura ed Applicata","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10231-024-01429-z","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 article, we investigate some new asymptotic behaviors of the solution for a two-component b-family equations. We first prove the persistence properties of the solution for Eq. (1.1) when the initial data decay logarithmically, algebraically at infinity with the power \(\beta \in (0,\infty )\). Subsequently, we obtain the infinite propagation of the solution to Eq. (1.1). If the initial data satisfy certain compact condition, then the nontrivial solution u of Eq. (1.1) immediately loses compactly supported. Meanwhile, the solution u decays exponentially as \(|x|\rightarrow \infty \).
在本文中,我们研究了双分量 b 族方程解的一些新的渐近行为。我们首先证明了方程(1.1)的解在初始数据对数衰减时的持久性,代数上在无穷大处有 \(\beta \in (0,\infty )\) 的幂。随后,我们得到式(1.1)解的无限传播。如果初始数据满足一定的紧凑条件,那么式(1.1)的非琐解 u 就会立即失去紧凑支撑。同时,解 u 会以指数形式衰减(|x|\rightarrow \infty \)。
期刊介绍:
This journal, the oldest scientific periodical in Italy, was originally edited by Barnaba Tortolini and Francesco Brioschi and has appeared since 1850. Nowadays it is managed by a nonprofit organization, the Fondazione Annali di Matematica Pura ed Applicata, c.o. Dipartimento di Matematica "U. Dini", viale Morgagni 67A, 50134 Firenze, Italy, e-mail annali@math.unifi.it).
A board of Italian university professors governs the Fondazione and appoints the editors of the journal, whose responsibility it is to supervise the refereeing process. The names of governors and editors appear on the front page of each issue. Their addresses appear in the title pages of each issue.