{"title":"随时间变化的能量的广义梯度流及其在涉及可变指数的 PDE 中的应用","authors":"Goro Akagi, Naoki Tanaka","doi":"10.1007/s00030-024-00955-2","DOIUrl":null,"url":null,"abstract":"<p>The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.\n</p>","PeriodicalId":501665,"journal":{"name":"Nonlinear Differential Equations and Applications (NoDEA)","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generalized gradient flows for time-dependent energies and applications to PDEs involving variable exponents\",\"authors\":\"Goro Akagi, Naoki Tanaka\",\"doi\":\"10.1007/s00030-024-00955-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.\\n</p>\",\"PeriodicalId\":501665,\"journal\":{\"name\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Differential Equations and Applications (NoDEA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00030-024-00955-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Differential Equations and Applications (NoDEA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00030-024-00955-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Generalized gradient flows for time-dependent energies and applications to PDEs involving variable exponents
The present paper presents an abstract theory for proving (local-in-time) existence of energy solutions to some doubly-nonlinear evolution equations of gradient flow type involving time-dependent subdifferential operators with a quantitative estimate for the local-existence time. Furthermore, the abstract theory is employed to obtain an optimal existence result for some doubly-nonlinear parabolic equations involving space-time variable exponents, which are (possibly) non-monotone in time. More precisely, global-in-time existence of solutions is proved for the parabolic equations.