{"title":"Global Existence and Decay Property for the Cauchy Problem of the Nonlinear MGT Plate Equation","authors":"Danhua Wang, Wenjun Liu","doi":"10.1007/s00245-024-10126-5","DOIUrl":null,"url":null,"abstract":"<div><p>We study the asymptotic behavior of the nonlinear MGT plate equation in the unbounded domain. By using semigroup theory, we first establish the well-posedness result for the Cauchy problem related to the linear MGT plate equation. By using the energy method in the Fourier space, we then prove the optimal decay estimate results for the non-critical case, in which the optimality is analyzed by considering the asymptotic expansion of the eigenvalues. By using the contraction mapping, we also show the local existence for the Cauchy problem of the nonlinear plate in appropriate function spaces, based on which we prove a global existence result for small data by using a priori energy estimates. Finally, based on the decay estimation of linear problems, the decay results of nonlinear problems are obtained.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"89 2","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10126-5","RegionNum":2,"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
We study the asymptotic behavior of the nonlinear MGT plate equation in the unbounded domain. By using semigroup theory, we first establish the well-posedness result for the Cauchy problem related to the linear MGT plate equation. By using the energy method in the Fourier space, we then prove the optimal decay estimate results for the non-critical case, in which the optimality is analyzed by considering the asymptotic expansion of the eigenvalues. By using the contraction mapping, we also show the local existence for the Cauchy problem of the nonlinear plate in appropriate function spaces, based on which we prove a global existence result for small data by using a priori energy estimates. Finally, based on the decay estimation of linear problems, the decay results of nonlinear problems are obtained.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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