{"title":"On the asymptotic behavior of the energy for evolution models with oscillating time-dependent damping","authors":"Halit Sevki Aslan, Marcelo Rempel Ebert","doi":"10.3233/asy-231851","DOIUrl":null,"url":null,"abstract":"In the present paper, we study the influence of oscillations of the time-dependent damping term b ( t ) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.","PeriodicalId":55438,"journal":{"name":"Asymptotic Analysis","volume":null,"pages":null},"PeriodicalIF":1.1000,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Asymptotic Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3233/asy-231851","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we study the influence of oscillations of the time-dependent damping term b ( t ) u t on the asymptotic behavior of the energy for solutions to the Cauchy problem for a σ-evolution equation u t t + ( − Δ ) σ u + b ( t ) u t = 0 , ( t , x ) ∈ [ 0 , ∞ ) × R n , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ R n , where σ > 0 and b is a continuous and positive function. Mainly we consider damping terms that are perturbations of the scale invariant case b ( t ) = β ( 1 + t ) − 1 , with β > 0, and we discuss the influence of oscillations of b on the energy estimates according to the size of β.
在本文中,我们研究的影响时间的振荡阻尼项b (t) u t解的渐近性态的能量σ进化论方程的柯西问题u t t +(−Δ)σu + b (t) t = 0时,(t, x)∈(0,∞)×R n, u (0, x) = 0 (x), u t (0, x) = 1 (x), x∈R n,其中σ> 0和b是一个持续的和积极的作用。我们主要考虑的阻尼项是尺度不变情况b (t) = β (1 + t)−1，β > 0的扰动，并根据β的大小讨论了b的振荡对能量估计的影响。
期刊介绍：
The journal Asymptotic Analysis fulfills a twofold function. It aims at publishing original mathematical results in the asymptotic theory of problems affected by the presence of small or large parameters on the one hand, and at giving specific indications of their possible applications to different fields of natural sciences on the other hand. Asymptotic Analysis thus provides mathematicians with a concentrated source of newly acquired information which they may need in the analysis of asymptotic problems.
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