{"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":"45 1","pages":""},"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":"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}
. 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":"307 1","pages":""},"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":"31 1","pages":""},"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":"46 1","pages":""},"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}
. In this paper, we investigate the initial value problem for the two-dimensional magneto-micropolar fluid equations with partial viscosity. We prove that global existence of smooth large solutions by the energy method. Furthermore, with aid of the Fourier splitting methods, optimal time-decay rates of global smooth large solutions are also established.
{"title":"Global Existence and Time-decay Rates of Solutions to 2D Magneto-micropolar Fluid Equations with Partial Viscosity","authors":"Yuzhu Wang null, Yuzhu Wang","doi":"10.4208/jpde.v35.n2.5","DOIUrl":"https://doi.org/10.4208/jpde.v35.n2.5","url":null,"abstract":". In this paper, we investigate the initial value problem for the two-dimensional magneto-micropolar fluid equations with partial viscosity. We prove that global existence of smooth large solutions by the energy method. Furthermore, with aid of the Fourier splitting methods, optimal time-decay rates of global smooth large solutions are also established.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"21 2 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83139525","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 obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R 2 . This ex-tends recent results of Hou (J. Inequal. Appl., 2018) and similar result was proved by Zhu (Sci. China Math., 2021).
{"title":"A Weighted Singular Trudinger-Moser Inequality","authors":"YU Peng","doi":"10.4208/jpde.v35.n3.2","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.2","url":null,"abstract":". In this paper, we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R 2 . This ex-tends recent results of Hou (J. Inequal. Appl., 2018) and similar result was proved by Zhu (Sci. China Math., 2021).","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"140 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77594143","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}
Rui Lu, Chunxiao Guo, Xin-Guang Yang null, P. Zhang
{"title":"Dynamics for Three Dimensional Generalized Navier- Stokes Equations with Delay","authors":"Rui Lu, Chunxiao Guo, Xin-Guang Yang null, P. Zhang","doi":"10.4208/jpde.v35.n2.2","DOIUrl":"https://doi.org/10.4208/jpde.v35.n2.2","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"14 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75252164","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 have discussed solution and stability analysis of ordinary and partial differential equation with boundary value problem. We investigated pe-riodic stability in Eulers scheme and also discussed PDEs by finite difference scheme. Numerical example has been discussed finding nature of stability. All given result more accurate other than existing methods.
{"title":"Study of Stability Criteria of Numerical Solution of Ordinary and Partial Differential Equations Using Euler’s and Finite Difference Scheme","authors":"Najmuddin Ahmad null, Shiv Charan","doi":"10.4208/jpde.v35.n3.6","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.6","url":null,"abstract":". In this paper we have discussed solution and stability analysis of ordinary and partial differential equation with boundary value problem. We investigated pe-riodic stability in Eulers scheme and also discussed PDEs by finite difference scheme. Numerical example has been discussed finding nature of stability. All given result more accurate other than existing methods.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"24 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81716531","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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