{"title":"Marcinkiewicz type operators with kernels involving Bessel functions","authors":"Laith Hawawsheh, Ahmad Al-Salman","doi":"10.3934/cpaa.2023086","DOIUrl":"https://doi.org/10.3934/cpaa.2023086","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Asymptotic behavior for wave equations with space-dependent damping in a weighted energy class","authors":"M. Sobajima","doi":"10.3934/cpaa.2023058","DOIUrl":"https://doi.org/10.3934/cpaa.2023058","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Random attractors for non-autonomous stochastic Navier-Stokes-Voigt equations in some unbounded domains","authors":"Shu Wang, Mengmeng Si, Rong Yang","doi":"10.3934/cpaa.2023062","DOIUrl":"https://doi.org/10.3934/cpaa.2023062","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Singular limits of invariant measures of the 3D MHD-Voigt equations","authors":"Yuanyuan Zhang, Guanggan Chen","doi":"10.3934/cpaa.2023116","DOIUrl":"https://doi.org/10.3934/cpaa.2023116","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135312070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In dimension $ Ngeq 5 $, and for $ 0< s<4 $ with $ gammain mathbb{R} $, we study the existence of nontrivial weak solutions for the doubly critical problem$ Delta^2 u-frac{gamma}{|x|^4}u = |u|^{2^star_0-2}u+frac{|u|^{ 2_s^{star}-2}u}{|x|^s}hbox{ in } mathbb{R}_+^N, ; u = Delta u = 0hbox{ on }partial mathbb{R}_+^N, $where $ 2_s^{star}: = frac{2(N-s)}{N-4} $ is the critical Hardy–Sobolev exponent. For $ Ngeq 8 $ and $ 0< gamma
在$ Ngeq 5 $维，对于$ 0< s<4 $和$ gammain mathbb{R} $，我们研究了双临界问题$ Delta^2 u-frac{gamma}{|x|^4}u = |u|^{2^star_0-2}u+frac{|u|^{ 2_s^{star}-2}u}{|x|^s}hbox{ in } mathbb{R}_+^N, ; u = Delta u = 0hbox{ on }partial mathbb{R}_+^N, $的非平凡弱解的存在性，其中$ 2_s^{star}: = frac{2(N-s)}{N-4} $是临界Hardy-Sobolev指数。对于$ Ngeq 8 $和$ 0< gamma<frac{(N^2-4)^2}{16} $，我们利用Ambrosetti-Rabinowitz的Mountain-Pass定理证明了非平凡解的存在性。所使用的方法是基于我们在文中证明的某些Hardy-Sobolev嵌入的极值的存在性。
{"title":"Fourth order Hardy-Sobolev equations: Singularity and doubly critical exponent","authors":"Hussein Cheikh Ali","doi":"10.3934/cpaa.2023112","DOIUrl":"https://doi.org/10.3934/cpaa.2023112","url":null,"abstract":"In dimension $ Ngeq 5 $, and for $ 0< s<4 $ with $ gammain mathbb{R} $, we study the existence of nontrivial weak solutions for the doubly critical problem$ Delta^2 u-frac{gamma}{|x|^4}u = |u|^{2^star_0-2}u+frac{|u|^{ 2_s^{star}-2}u}{|x|^s}hbox{ in } mathbb{R}_+^N, ; u = Delta u = 0hbox{ on }partial mathbb{R}_+^N, $where $ 2_s^{star}: = frac{2(N-s)}{N-4} $ is the critical Hardy–Sobolev exponent. For $ Ngeq 8 $ and $ 0< gamma<frac{(N^2-4)^2}{16} $, we show the existence of nontrivial solution using the Mountain-Pass theorem by Ambrosetti-Rabinowitz. The method used is based on the existence of extremals for certain Hardy-Sobolev embeddings that we prove in this paper.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136366896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the weighted Dirichlet eigenvalues of Hardy operators involving critical gradient terms","authors":"Ying Wang, Yanjing Qiu, Qingping Yin","doi":"10.3934/cpaa.2023023","DOIUrl":"https://doi.org/10.3934/cpaa.2023023","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Coexistence of hyperbolic and elliptic invariant tori for completely degenerate quasi-periodically forced maps","authors":"Guangzhao Zhou, Yehui Zhang, Wen Si","doi":"10.3934/cpaa.2023029","DOIUrl":"https://doi.org/10.3934/cpaa.2023029","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the well-posedness and stability for a coupled nonlinear suspension bridge problem","authors":"S. E. Mukiawa, M. Leblouba, S. Messaoudi","doi":"10.3934/cpaa.2023084","DOIUrl":"https://doi.org/10.3934/cpaa.2023084","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem $ begin{equation} Lu = |u|^{p-2}u+mu |u|^{q-2}uquadtext{in}; ; Omega, quadquad u = 0quadtext{in}; ; mathbb{R}^NsetminusOmega, ~~~(1) end{equation} $ where $ L: = -Delta +(-Delta)^s $ for $ sin(0, 1) $ and $ Omegasubsetmathbb{R}^N $ is a bounded domain. Precisely, we show that problem (1) for $ 1
本文研究了一类具有超临界非线性的混合局部和非局部狄利克雷问题。我们首先建立了问题$ begin{equation} Lu = |u|^{p-2}u+mu |u|^{q-2}uquadtext{in}; ; Omega, quadquad u = 0quadtext{in}; ; mathbb{R}^NsetminusOmega, ~~~(1) end{equation} $的多重性结果，其中$ sin(0, 1) $和$ Omegasubsetmathbb{R}^N $的$ L: = -Delta +(-Delta)^s $是一个有界域。准确地说，我们证明了$ 1<q<2<p $的问题(1)有一个正解，以及对于$ mu $的小值具有负能量的换号解序列。这里$ u $可以是标量函数，也可以是矢量值函数，这样(1)就变成了一个具有超临界非线性的系统。此外，只要定义域是对称的，我们也证明了具有相同对称性的对称解的存在性。我们还证明了在$ Omega $上具有Dirichlet边界条件的超临界哈密顿系统$ Lu = |v|^{p-2}v, qquad Lv = |u|^{d-2}u+mu |u|^{q-2}u $的存在性结果，其中$ 1<q<2<p, d $。我们的方法是变分的，并且在这两个问题中，超临界问题的紧性缺失通过在适当函数空间的闭凸子集上工作来恢复。
{"title":"Mixed local and nonlocal supercritical Dirichlet problems","authors":"David Amundsen, Abbas Moameni, Remi Yvant Temgoua","doi":"10.3934/cpaa.2023104","DOIUrl":"https://doi.org/10.3934/cpaa.2023104","url":null,"abstract":"In this work, we consider a mixed local and nonlocal Dirichlet problem with supercritical nonlinearity. We first establish a multiplicity result for the problem $ begin{equation} Lu = |u|^{p-2}u+mu |u|^{q-2}uquadtext{in}; ; Omega, quadquad u = 0quadtext{in}; ; mathbb{R}^NsetminusOmega, ~~~(1) end{equation} $ where $ L: = -Delta +(-Delta)^s $ for $ sin(0, 1) $ and $ Omegasubsetmathbb{R}^N $ is a bounded domain. Precisely, we show that problem (1) for $ 1<q<2<p $ has a positive solution as well as a sequence of sign-changing solutions with a negative energy for small values of $ mu $. Here $ u $ can be either a scalar function, or a vector valued function so that (1) turns into a system with supercritical nonlinearity. Moreover, whenever the domain is symmetric, we also prove the existence of symmetric solutions enjoying the same symmetry properties. We shall also prove an existence result for the supercritical Hamiltonian system$ Lu = |v|^{p-2}v, qquad Lv = |u|^{d-2}u+mu |u|^{q-2}u $with the Dirichlet boundary condition on $ Omega $ where $ 1<q<2<p, d $. Our method is variational, and in both problems the lack of compactness for the supercritical problem is recovered by working on a closed convex subset of an appropriate function space.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135496197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the long-time behavior of scalar viscous conservation laws via the structure of $ omega $-limit sets. We show that $ omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.
{"title":"Viscous shocks and long-time behavior of scalar conservation laws","authors":"Thierry Gallay, Arnd Scheel","doi":"10.3934/cpaa.2023119","DOIUrl":"https://doi.org/10.3934/cpaa.2023119","url":null,"abstract":"We study the long-time behavior of scalar viscous conservation laws via the structure of $ omega $-limit sets. We show that $ omega $-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $ omega $-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135612093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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