{"title":"On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms","authors":"Marco Discacciati, Claudia Garetto, Costas Loizou","doi":"10.1007/s10440-023-00601-6","DOIUrl":null,"url":null,"abstract":"<div><p>This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in <i>J. Differ. Equ.</i> <b>319</b> (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in <i>Arch. Ration. Mech. Anal.</i> <b>217</b> (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":"187 1","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00601-6.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Applicandae Mathematicae","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10440-023-00601-6","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
This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.
在时间相关系数的情况下，本文补充了（Discacciati et al.in J.Differ.Equ.319（2022）131–185）中开始的具有不连续系数的波动方程的研究。在这里，我们假设方程系数仅取决于空间，并且我们在低阶项上公式化Levi条件，以保证存在如（Garetto和Ruzhansky in Arch.Ration.Mech.Anal.217（2015）113–154）中定义的非常弱的解。作为一个玩具模型，我们研究了具有不连续速度的保守形式的波动方程，并通过数值方法对相应的非常弱的解进行了定性分析。
期刊介绍：
Acta Applicandae Mathematicae is devoted to the art and techniques of applying mathematics and the development of new, applicable mathematical methods.
Covering a large spectrum from modeling to qualitative analysis and computational methods, Acta Applicandae Mathematicae contains papers on different aspects of the relationship between theory and applications, ranging from descriptive papers on actual applications meeting contemporary mathematical standards to proofs of new and deep theorems in applied mathematics.
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