{"title":"Blowup Behavior of Solutions to an $omega$-diffusion Equation on the Graph","authors":"Liping Zhu null, Lin Huang","doi":"10.4208/jpde.v35.n2.3","DOIUrl":"https://doi.org/10.4208/jpde.v35.n2.3","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"176 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76227262","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 a class of sublinear operators and their commutators with a weighted BMO function. We first give the definition of a weighted Morrey space L p , κ µ , ω ( X ) where X is an RD-measure and ω is the weight function. The weighted Morrey spaces arise from studying the local behavior of solutions to certain partial differential equations. We will show that the aforementioned class of operators and their communtators with a weighted BMO function are bounded in the weighted Morrey space L p , κ µ , ω ( X ) provided that the weight function ω belongs to the A p ( µ ) -class and satisfies the reverse H¨older’s condition.
{"title":"Boundedness for a Class of Operators on Weighted Morrey Space with RD-measure","authors":"Xiaona Cui, Yongjin Lu null, Mengmeng Li","doi":"10.4208/jpde.v35.n4.7","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.7","url":null,"abstract":". In this paper, we study a class of sublinear operators and their commutators with a weighted BMO function. We first give the definition of a weighted Morrey space L p , κ µ , ω ( X ) where X is an RD-measure and ω is the weight function. The weighted Morrey spaces arise from studying the local behavior of solutions to certain partial differential equations. We will show that the aforementioned class of operators and their communtators with a weighted BMO function are bounded in the weighted Morrey space L p , κ µ , ω ( X ) provided that the weight function ω belongs to the A p ( µ ) -class and satisfies the reverse H¨older’s condition.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"56 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74043010","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":"Exact Boundary Controllability of Fifth-order KdV Equation Posed on the Periodic Domain","authors":"Shuning Yang null, Xiangqing Zhao","doi":"10.4208/jpde.v35.n2.4","DOIUrl":"https://doi.org/10.4208/jpde.v35.n2.4","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"30 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82992504","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 are interested in the following nonlocal problem with critical exponent where a , b are positive constants, 2 < p < 6, Ω is a smooth bounded domain in R 3 and λ > 0 is a parameter. By variational methods, we prove that problem has a positive ground state solution u b for λ > 0 sufficiently large. Moreover, we take b as a parameter and study the asymptotic behavior of u b when b ց 0.
{"title":"Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three","authors":"X. Qian","doi":"10.4208/jpde.v35.n4.6","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.6","url":null,"abstract":". In this paper, we are interested in the following nonlocal problem with critical exponent where a , b are positive constants, 2 < p < 6, Ω is a smooth bounded domain in R 3 and λ > 0 is a parameter. By variational methods, we prove that problem has a positive ground state solution u b for λ > 0 sufficiently large. Moreover, we take b as a parameter and study the asymptotic behavior of u b when b ց 0.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"47 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":"83189563","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":"On Local Well posedness of the Schrödinger-Boussinesq Systems","authors":"N. Null","doi":"10.4208/jpde.v35.n4.5","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.5","url":null,"abstract":"","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":" 30","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72499577","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 ( X , d , µ ) be a metric space with a Borel-measure µ , suppose µ satisfies the Ahlfors-regular condition, i.e. where b 1 , b 2 are two positive constants and s is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.
{"title":"A Singular Moser-Trudinger Inequality on Metric Measure Space","authors":"Yaoting Gui","doi":"10.4208/jpde.v35.n4.3","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.3","url":null,"abstract":". Let ( X , d , µ ) be a metric space with a Borel-measure µ , suppose µ satisfies the Ahlfors-regular condition, i.e. where b 1 , b 2 are two positive constants and s is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that s is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"154 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86043312","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, we study the multiplicity of solutions for the fractional Schr¨odinger equation with s ∈ ( 0,1 ) , N ≥ 3, p ∈ ( 1, 2 N N − 2 s − 1 ) and lim | y |→ + ∞ V ( y ) > 0. By assuming suitable decay property of the radial potential V ( y ) = V ( | y | ) , we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator ( − ∆ ) s and the algebraic decay involving the approximation solutions make the estimates become more subtle.
． 本文研究了s∈(0,1)，N≥3,p∈(1,2 N N−2 s−1)，lim | y |→+∞V (y) > 0的分数阶Schr¨odinger方程解的多重性。通过假设径向势V (y) = V (| y |)具有合适的衰减性质，构造了另一类集中于两个边长无限大的类似等边多边形无穷顶点处的解。因此，除了每个多边形的长度外，我们还必须同时考虑另一个参数，即足架的高度。另一个困难在于算子(−∆)s的非局部性质，以及涉及近似解的代数衰减使估计变得更加微妙。
{"title":"Infinitely Many Solutions for the Fractional Nonlinear Schrödinger Equations of a New Type","authors":"Qing Guo null, Lixiu Duan","doi":"10.4208/jpde.v35.n3.5","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.5","url":null,"abstract":". This paper, we study the multiplicity of solutions for the fractional Schr¨odinger equation with s ∈ ( 0,1 ) , N ≥ 3, p ∈ ( 1, 2 N N − 2 s − 1 ) and lim | y |→ + ∞ V ( y ) > 0. By assuming suitable decay property of the radial potential V ( y ) = V ( | y | ) , we construct another type of solutions concentrating at infinite vertices of two similar equilateral polygonal with infinitely large length of sides. Hence, besides the length of each polygonal, we must consider one more parameter, that is the height of the podetium, simultaneously. Another difficulty lies in the non-local property of the operator ( − ∆ ) s and the algebraic decay involving the approximation solutions make the estimates become more subtle.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"56 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83973831","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 existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs. Our method is based on calculus of variations, which was built on connected finite graphs by Grigor’yan, Lin and Yang.
{"title":"Mean Field Equations for the Equilibrium Turbulence and Toda Systems on Connected Finite Graphs","authors":"Xiao-Dan Zhu","doi":"10.4208/jpde.v35.n3.1","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.1","url":null,"abstract":". In this paper, we study existence of solutions of mean field equations for the equilibrium turbulence and Toda systems on connected finite graphs. Our method is based on calculus of variations, which was built on connected finite graphs by Grigor’yan, Lin and Yang.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"28 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74446690","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 two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the global existence and uniqueness of solutions to 2D Prandtl-Hartmann equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann equations can be extended up to T ε (see Theorem 2.1) when the strength of the perturbation is of the order of ε . The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of x -derivative in the term v ∂ y u . To overcome this difficulty, we introduce the Gaussian weighted Poincar´ e inequality (see Lemma 2.3). Com-pared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework, either.
{"title":"Global Well-Posedness of Solutions to 2D Prandtl-Hartmann Equations in Analytic Framework","authors":"Xiaolei Dong null, Yuming Qin","doi":"10.4208/jpde.v35.n3.7","DOIUrl":"https://doi.org/10.4208/jpde.v35.n3.7","url":null,"abstract":". In this paper, we consider the two-dimensional (2D) Prandtl-Hartmann equations on the half plane and prove the global existence and uniqueness of solutions to 2D Prandtl-Hartmann equations by using the classical energy methods in analytic framework. We prove that the lifespan of the solutions to 2D Prandtl-Hartmann equations can be extended up to T ε (see Theorem 2.1) when the strength of the perturbation is of the order of ε . The difficulty of solving the Prandtl-Hartmann equations in the analytic framework is the loss of x -derivative in the term v ∂ y u . To overcome this difficulty, we introduce the Gaussian weighted Poincar´ e inequality (see Lemma 2.3). Com-pared to the existence and uniqueness of solutions to the classical Prandtl equations where the monotonicity condition of the tangential velocity plays a key role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework. Besides, the existence and uniqueness of solutions to the 2D MHD boundary layer where the initial tangential magnetic field has a lower bound plays an important role, which is not needed for the 2D Prandtl-Hartmann equations in analytic framework, either.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"16 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74810291","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 Cauchy problem of 3D tropical climate model with zero thermal diffusion. Firstly, we establish the global regularity for this system with fractional diffusion α = β = 5/4. Secondly, by adding only a damp term, we obtain the global well-posedness for small initial data.
{"title":"Global Well-Posedness for the 3D Tropical Climate Model without Thermal Diffusion","authors":"Yanghai Yu, Jinlu Li null, Xiuwei Yin","doi":"10.4208/jpde.v35.n4.4","DOIUrl":"https://doi.org/10.4208/jpde.v35.n4.4","url":null,"abstract":". In this paper, we consider the Cauchy problem of 3D tropical climate model with zero thermal diffusion. Firstly, we establish the global regularity for this system with fractional diffusion α = β = 5/4. Secondly, by adding only a damp term, we obtain the global well-posedness for small initial data.","PeriodicalId":43504,"journal":{"name":"Journal of Partial Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89539675","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：