{"title":"非局部非线性抛物问题弱解的存在性、唯一性和连续性结果","authors":"Tayeb Benhamoud, E. Zaouche, M. Bousselsal","doi":"10.21136/mb.2024.0065-23","DOIUrl":null,"url":null,"abstract":". This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation u t − M ( R Ω φu d x )div ( A ( x, t, u ) ∇ u ) = g ( x, t, u ) in Ω × (0 , T ), where Ω is a bounded domain of R n ( n > 1), T > 0 is a positive number, A ( x, t, u ) is an n × n matrix of variable coefficients depending on u and M : R → R , φ : Ω → R , g : Ω × (0 , T ) × R → R are given functions. We consider two different assumptions on g . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if A ( x, t, u ) = a ( x, t ) depends only on the variable ( x, t ), we investigate two uniqueness theorems and give a continuity result depending on the initial data.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems\",\"authors\":\"Tayeb Benhamoud, E. Zaouche, M. Bousselsal\",\"doi\":\"10.21136/mb.2024.0065-23\",\"DOIUrl\":null,\"url\":null,\"abstract\":\". This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation u t − M ( R Ω φu d x )div ( A ( x, t, u ) ∇ u ) = g ( x, t, u ) in Ω × (0 , T ), where Ω is a bounded domain of R n ( n > 1), T > 0 is a positive number, A ( x, t, u ) is an n × n matrix of variable coefficients depending on u and M : R → R , φ : Ω → R , g : Ω × (0 , T ) × R → R are given functions. We consider two different assumptions on g . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if A ( x, t, u ) = a ( x, t ) depends only on the variable ( x, t ), we investigate two uniqueness theorems and give a continuity result depending on the initial data.\",\"PeriodicalId\":45392,\"journal\":{\"name\":\"Mathematica Bohemica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2024-02-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Bohemica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.21136/mb.2024.0065-23\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Bohemica","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.21136/mb.2024.0065-23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文主要研究与方程 u t - M ( R Ω φu d x ) div ( A ( x, t, u ) ∇ u ) = g ( x, t, u ) 在 Ω × (0 , T ) 中相关的非局部非线性抛物问题,其中 Ω 是 R n ( n > 1) 的有界域,T > 0 是一个正数,A ( x, t, u ) 是一个 n × n 的可变系数矩阵,取决于 u 和 M:M : R → R , φ : Ω → R , g : Ω × (0 , T ) × R → R 是给定函数。我们考虑对 g 的两种不同假设。对于这两种假设,我们都可以利用绍德定点定理证明这个问题存在弱解。此外,如果 A ( x, t, u ) = a ( x, t ) 只取决于变量 ( x, t ) ,我们研究了两个唯一性定理,并给出了一个取决于初始数据的连续性结果。
Existence, uniqueness and continuity results of weak solutions for nonlocal nonlinear parabolic problems
. This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation u t − M ( R Ω φu d x )div ( A ( x, t, u ) ∇ u ) = g ( x, t, u ) in Ω × (0 , T ), where Ω is a bounded domain of R n ( n > 1), T > 0 is a positive number, A ( x, t, u ) is an n × n matrix of variable coefficients depending on u and M : R → R , φ : Ω → R , g : Ω × (0 , T ) × R → R are given functions. We consider two different assumptions on g . The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if A ( x, t, u ) = a ( x, t ) depends only on the variable ( x, t ), we investigate two uniqueness theorems and give a continuity result depending on the initial data.