{"title":"Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity","authors":"Dhouha Draouil, Mohamed Majdoub","doi":"10.1007/s43034-025-00427-3","DOIUrl":null,"url":null,"abstract":"<div><p>We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser–Trudinger-type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Furthermore, we derive the linearization of energy-bounded solutions using the methodology introduced in Gérard (J Funct Anal 141:60–98, 1996). The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty of this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.</p></div>","PeriodicalId":48858,"journal":{"name":"Annals of Functional Analysis","volume":"16 3","pages":""},"PeriodicalIF":1.2000,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s43034-025-00427-3","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
We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser–Trudinger-type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Furthermore, we derive the linearization of energy-bounded solutions using the methodology introduced in Gérard (J Funct Anal 141:60–98, 1996). The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty of this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.
期刊介绍：
Annals of Functional Analysis is published by Birkhäuser on behalf of the Tusi Mathematical Research Group.
Ann. Funct. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and all modern related topics (e.g., operator theory). Ann. Funct. Anal. normally publishes original research papers numbering 18 or fewer pages in the journal’s style. Longer papers may be submitted to the Banach Journal of Mathematical Analysis or Advances in Operator Theory.
Ann. Funct. Anal. presents the best paper award yearly. The award in the year n is given to the best paper published in the years n-1 and n-2. The referee committee consists of selected editors of the journal.
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