. Consider ( M , g ) as an n -dimensional compact Riemannian manifold. In this paper we are going to study a class of elliptic differential operators which appears naturally in the study of hypersurfaces with constant mean curvature and also the study of variation theory for 1-area functional.
{"title":"The Eigenvalues of a Class of Elliptic Differential Operators","authors":"Mohammad Javad Habibi Vosta Kolaei null, S. Azami","doi":"10.4208/jpde.v36.n1.4","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.4","url":null,"abstract":". Consider ( M , g ) as an n -dimensional compact Riemannian manifold. In this paper we are going to study a class of elliptic differential operators which appears naturally in the study of hypersurfaces with constant mean curvature and also the study of variation theory for 1-area functional.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"112 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89040201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Blow-Up and Boundedness in Quasilinear Parabolic-Elliptic Chemotaxis System with Nonlinear Signal Production","authors":"Ruxi Cao null, Zhongping Li","doi":"10.4208/jpde.v36.n3.2","DOIUrl":"https://doi.org/10.4208/jpde.v36.n3.2","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"74 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79475035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we mainly investigate the initial boundary value problem for a plate equation with non-local degenerate energy damping term and source term. By using potential well and Nakao’s inequality, we establish the global existence and the energy decay rate when the initial data is starting in the stable set. Finally, we derive some further estimate on the stability.
{"title":"Energy Decay for a Type of Plate Equation with Degenerate Energy Damping and Source Term","authors":"Gongwei Liu, Yang null, Hongwei Zhang","doi":"10.4208/jpde.v36.n3.4","DOIUrl":"https://doi.org/10.4208/jpde.v36.n3.4","url":null,"abstract":". In this paper, we mainly investigate the initial boundary value problem for a plate equation with non-local degenerate energy damping term and source term. By using potential well and Nakao’s inequality, we establish the global existence and the energy decay rate when the initial data is starting in the stable set. Finally, we derive some further estimate on the stability.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"7 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73438262","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jina Li, Gai-zhu Qu, Jianlin Zhang null, Xuehui Ji
. In this research, invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method, and the generalized second-order Kudryashov-Sinelshchikov equation is used to describe pressure waves in a liquid with bubbles. The governing equations are classified and transformed into a system of ordinary differential equations, and the exact solutions of the classified equation are obtained by solving the system of ordinary differential equations. The method gives logarithmic, polynomial, exponential, and trigonometric solutions for equations. The primary accomplishments of this work are displaying how to obtain the exact solutions of the classified equations and giving the stability analysis of reduced ordinary differential equations.
{"title":"Invariant Subspace and Exact Solutions to the Generalized Kudryashov-Sinelshchikov Equation","authors":"Jina Li, Gai-zhu Qu, Jianlin Zhang null, Xuehui Ji","doi":"10.4208/jpde.v36.n3.3","DOIUrl":"https://doi.org/10.4208/jpde.v36.n3.3","url":null,"abstract":". In this research, invariant subspaces and exact solutions for the governing equation are obtained through the invariant subspace method, and the generalized second-order Kudryashov-Sinelshchikov equation is used to describe pressure waves in a liquid with bubbles. The governing equations are classified and transformed into a system of ordinary differential equations, and the exact solutions of the classified equation are obtained by solving the system of ordinary differential equations. The method gives logarithmic, polynomial, exponential, and trigonometric solutions for equations. The primary accomplishments of this work are displaying how to obtain the exact solutions of the classified equations and giving the stability analysis of reduced ordinary differential equations.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"74 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76128756","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we consider the following Schr¨odinger-Poisson system
． 在本文中，我们考虑如下Schr¨odinger-Poisson系统
{"title":"Positive Ground State Solutions for Schrödinger-Poisson System with General Nonlinearity and Critical Exponent","authors":"Qingfang Chen null, Jia‐Feng Liao","doi":"10.4208/jpde.v36.n1.5","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.5","url":null,"abstract":". In this paper, we consider the following Schr¨odinger-Poisson system","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"25 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81077927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Let W 1, n ( R n ) be the standard Sobolev space. For any τ > 0 and p > n > 2, we denote
{"title":"Extremal Functions for an Improved Trudinger-Moser Inequality Involving $L^p$-Norm in $mathbb{R}^n$","authors":"YANG Liu null, Xiaomeng LI","doi":"10.4208/jpde.v36.n4.7","DOIUrl":"https://doi.org/10.4208/jpde.v36.n4.7","url":null,"abstract":". Let W 1, n ( R n ) be the standard Sobolev space. For any τ > 0 and p > n > 2, we denote","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension n = 1 and the nonlinear power is bigger than 2, the life-span (cid:101) T ( ε ) and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index K , which depends on the time-dependent damping and the nonlinear term, the life-span (cid:101) T ( ε ) can be estimated below by ε − p 1 − K , e ε − p or + ∞ , where ε is the scale of the compact support of the initial data.
． 研究一类具有时变阻尼的半线性波动方程的柯西问题。在空间维数n = 1且非线性幂大于2的情况下，统一地研究了该问题经典解的寿命(cid:101) T (ε)和整体存在性。更准确地说，对于依赖于时间相关阻尼和非线性项的指标K的不同值，寿命(cid:101) T (ε)可以用ε−p 1−K, e ε−p或+∞来估计，其中ε是初始数据的紧支持的尺度。
{"title":"Life-Span of Classical Solutions to a Semilinear Wave Equation with Time-Dependent Damping","authors":"F. Guo, Jinling Liang null, Changwang Xiao","doi":"10.4208/jpde.v36.n3.1","DOIUrl":"https://doi.org/10.4208/jpde.v36.n3.1","url":null,"abstract":". This paper is concerned with the Cauchy problem for a semilinear wave equation with a time-dependent damping. In case that the space dimension n = 1 and the nonlinear power is bigger than 2, the life-span (cid:101) T ( ε ) and global existence of the classical solution to the problem has been investigated in a unified way. More precisely, with respect to different values of an index K , which depends on the time-dependent damping and the nonlinear term, the life-span (cid:101) T ( ε ) can be estimated below by ε − p 1 − K , e ε − p or + ∞ , where ε is the scale of the compact support of the initial data.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"76 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83118841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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