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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
{"title":"Properties of Solutions to Fractional Laplace Equation with Singular Term","authors":"Wang Xinjing","doi":"10.4208/jpde.v36.n2.4","DOIUrl":"https://doi.org/10.4208/jpde.v36.n2.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77323672","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":null,"pages":null},"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}
{"title":"Free Boundaries Problem for a Class of Parabolic Type Chemotaxis Model","authors":"Wenbin LYU","doi":"10.4208/jpde.v36.n4.3","DOIUrl":"https://doi.org/10.4208/jpde.v36.n4.3","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135194310","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 study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists v a unique global-in-time solution for small initial data be-longing to the critical Fourier-Besov-Morrey spaces FN 3 − 2 α + λ − 2 p p , λ , q . Moreover, we show the asymptotic behavior of the global solution v . i.e., k v ( t ) k FN 3 − 2 α + λ − 2 p p , λ , q decays to zero as time goes to infinity.
． 本文研究了亚临界耗散准地转方程。利用Littlewood Paley理论、傅里叶分析和标准技术，证明了在临界傅里叶- besov - morrey空间FN 3−2 α + λ−2 p p， λ， q下存在唯一的全局实时解。此外，我们还证明了全局解v的渐近性质。即k v (t) k FN 3−2 α + λ−2 p p， λ， q随着时间趋于无穷衰减为零。
{"title":"Global Well-Posedness and Asymptotic Behavior for the 2D Subcritical Dissipative Quasi-Geostrophic Equation in Critical Fourier-Besov-Morrey Spaces","authors":"Achraf Azanzal, Chakir Allalou null, Adil Abbassi","doi":"10.4208/jpde.v36.n1.1","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.1","url":null,"abstract":". In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists v a unique global-in-time solution for small initial data be-longing to the critical Fourier-Besov-Morrey spaces FN 3 − 2 α + λ − 2 p p , λ , q . Moreover, we show the asymptotic behavior of the global solution v . i.e., k v ( t ) k FN 3 − 2 α + λ − 2 p p , λ , q decays to zero as time goes to infinity.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77061039","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 study the discrete Morse flow for either Yamabe type heat flow or nonlinear heat flow on a bounded regular domain in the whole space. We show that under suitable assumptions on the initial data g one has a weak approxi-mate discrete Morse flow for the Yamabe type heat flow on any time interval. This phenomenon is very different from the smooth Yamabe flow, where the finite time blow up may exist.
{"title":"Discrete Morse Flow for Yamabe Type Heat Flows","authors":"Ma Li null, Weiqiong Zheng","doi":"10.4208/jpde.v36.n1.3","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.3","url":null,"abstract":". In this paper, we study the discrete Morse flow for either Yamabe type heat flow or nonlinear heat flow on a bounded regular domain in the whole space. We show that under suitable assumptions on the initial data g one has a weak approxi-mate discrete Morse flow for the Yamabe type heat flow on any time interval. This phenomenon is very different from the smooth Yamabe flow, where the finite time blow up may exist.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75186888","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 divergence degenerate elliptic equation with bounded coefficients constructed by H¨ormander’s vector fields. We prove a De Giorgi type result, i.e., the local H¨older continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here. As a consequence, the Harnack inequality of weak solutions is also given
{"title":"A De Giorgi Type Result to Divergence Degenerate Elliptic Equation with Bounded Coefficients Related To Hörmander's Vector Fields","authors":"Lingling Hou","doi":"10.4208/jpde.v36.n1.2","DOIUrl":"https://doi.org/10.4208/jpde.v36.n1.2","url":null,"abstract":". In this paper, we consider the divergence degenerate elliptic equation with bounded coefficients constructed by H¨ormander’s vector fields. We prove a De Giorgi type result, i.e., the local H¨older continuity for the weak solutions to the equation by providing a De Giorgi type lemma and extending the Moser iteration to the setting here. As a consequence, the Harnack inequality of weak solutions is also given","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85388651","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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