{"title":"Quantitative estimates for space-time analyticity of solutions to the fractional Navier-Stokes equations","authors":"Cong Wang, Yu Gao, Xiaoping Xue","doi":"10.3934/cpaa.2023080","DOIUrl":"https://doi.org/10.3934/cpaa.2023080","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":"70221353","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":"Convergence rate for perturbations of Morse-Smale semiflow","authors":"L. Pires","doi":"10.3934/cpaa.2023082","DOIUrl":"https://doi.org/10.3934/cpaa.2023082","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":"70221427","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":"Interior interfaces with (or without) boundary intersection for an anisotropic Allen-Cahn equation","authors":"Suting Wei, Jun Yang","doi":"10.3934/cpaa.2023057","DOIUrl":"https://doi.org/10.3934/cpaa.2023057","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"171 3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221493","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 layer and stable solutions of nonlocal problem $ begin{equation*} -Delta u+F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{n} end{equation*} $ where $ Fin C_{{text{loc}}}^2( mathbb R) $ satisfies $ F(0) = 0 $ and $ G $ is a double well potential. For $ n = 2,s>0 $ and $ n = 3, $ $ sgeq 1/2, $ we establish the 1-d symmetry of layer solutions for this equation. When $ n = 2 $ and $ F' $ is bounded away from zero, we prove the 1-d symmetry of stable solutions for this equation. Using a different approach, we also prove the 1-d symmetry of stable solutions for$ begin{equation*} F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{2}. end{equation*} $
研究了非局部问题$ begin{equation*} -Delta u+F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{n} end{equation*} $的层解和稳定解，其中$ Fin C_{{text{loc}}}^2( mathbb R) $满足$ F(0) = 0 $, $ G $是双井势。对于$ n = 2,s>0 $和$ n = 3, $$ sgeq 1/2, $，我们建立了该方程层解的一维对称性。当$ n = 2 $和$ F' $离零有界时，证明了该方程稳定解的一维对称性。用一种不同的方法，证明了$ begin{equation*} F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{2}. end{equation*} $
{"title":"Layer and stable solutions to a nonlocal model","authors":"Xiaodong Yan","doi":"10.3934/cpaa.2023105","DOIUrl":"https://doi.org/10.3934/cpaa.2023105","url":null,"abstract":"We study the layer and stable solutions of nonlocal problem $ begin{equation*} -Delta u+F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{n} end{equation*} $ where $ Fin C_{{text{loc}}}^2( mathbb R) $ satisfies $ F(0) = 0 $ and $ G $ is a double well potential. For $ n = 2,s>0 $ and $ n = 3, $ $ sgeq 1/2, $ we establish the 1-d symmetry of layer solutions for this equation. When $ n = 2 $ and $ F' $ is bounded away from zero, we prove the 1-d symmetry of stable solutions for this equation. Using a different approach, we also prove the 1-d symmetry of stable solutions for$ begin{equation*} F'(u)left( -Delta right) ^{s}F(u)+G'(u) = 0text{ in }mathbb{R}^{2}. end{equation*} $","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"20 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":"135441599","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}
The well-posedness and asymptotic dynamics of non-autonomous wave equations with state-dependent delay and sup-cubic nonlinearity are investigated. Based on the Strichartz estimates, we first obtain the well-posedness in a $ C^1 $-type space. Then, we present a general scheme for considering the dynamics, which generalizes the method of quasi-stability to the non-autonomous setting. Applying this scheme to our concrete model, we establish the existence of a uniform attractor and give its entropy estimates.
{"title":"Uniform attractor and its Kolmogorov entropy for a damped sup-cubic wave equation with state-dependent delay","authors":"Yangmin Xiong, Xinyu Mei","doi":"10.3934/cpaa.2023115","DOIUrl":"https://doi.org/10.3934/cpaa.2023115","url":null,"abstract":"The well-posedness and asymptotic dynamics of non-autonomous wave equations with state-dependent delay and sup-cubic nonlinearity are investigated. Based on the Strichartz estimates, we first obtain the well-posedness in a $ C^1 $-type space. Then, we present a general scheme for considering the dynamics, which generalizes the method of quasi-stability to the non-autonomous setting. Applying this scheme to our concrete model, we establish the existence of a uniform attractor and give its entropy estimates.","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":"136366878","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 consider functionals of the form $ begin{equation*} mathcal{F}(u) = displaystyle{int}_{ Omega} f(x, u(x), D u(x)), dx, quad uin u_0 + W_0^{1, r}( Omega, {mathbb{R}^m}), end{equation*} $ where the integrand $ f = f(x, p, xi): Omegatimes mathbb{R}^mtimes mathbb{M}^{mtimes n} to mathbb{R} $ is assumed to be non-quasiconvex in the last variable and $ u_0 $ is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope $ overline{f} $ of $ f $ and of the relaxed functional$ begin{equation*} overline{mathcal{F}}(u) = displaystyle{int}_{ Omega} overline{f}(x, u(x), D u(x)), dx, quad uin u_0 + W_0^{1, r}( Omega, {mathbb{R}^m}), end{equation*} $imposing standard differentiability and growth properties on $ overline{f} $. Then we assume the quasiaffinity of $ overline{f} $ on the set in which $ f> overline{f} $ and the strict monotonicity of the map $ mathbb{R}ni p^i mapsto overline{f}(x, p, xi) $, where $ p^i $ is a single scalar component of the vector function variable $ p $, showing that any minimizer of $ overline{mathcal{F}} $ minimizes $ mathcal{F} $ too.
{"title":"Existence of minimizers for non-quasiconvex functionals by strict monotonicity","authors":"Sandro Zagatti","doi":"10.3934/cpaa.2023114","DOIUrl":"https://doi.org/10.3934/cpaa.2023114","url":null,"abstract":"We consider functionals of the form $ begin{equation*} mathcal{F}(u) = displaystyle{int}_{ Omega} f(x, u(x), D u(x)), dx, quad uin u_0 + W_0^{1, r}( Omega, {mathbb{R}^m}), end{equation*} $ where the integrand $ f = f(x, p, xi): Omegatimes mathbb{R}^mtimes mathbb{M}^{mtimes n} to mathbb{R} $ is assumed to be non-quasiconvex in the last variable and $ u_0 $ is an arbitrary boundary value. We study the minimum problem by the introduction of the lower quasiconvex envelope $ overline{f} $ of $ f $ and of the relaxed functional$ begin{equation*} overline{mathcal{F}}(u) = displaystyle{int}_{ Omega} overline{f}(x, u(x), D u(x)), dx, quad uin u_0 + W_0^{1, r}( Omega, {mathbb{R}^m}), end{equation*} $imposing standard differentiability and growth properties on $ overline{f} $. Then we assume the quasiaffinity of $ overline{f} $ on the set in which $ f> overline{f} $ and the strict monotonicity of the map $ mathbb{R}ni p^i mapsto overline{f}(x, p, xi) $, where $ p^i $ is a single scalar component of the vector function variable $ p $, showing that any minimizer of $ overline{mathcal{F}} $ minimizes $ mathcal{F} $ too.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"22 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":"136366909","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":"Entire subsolutions of a kind of $ boldsymbol{k} $-Hessian type equations with gradient terms","authors":"J. Ji, F. Jiang, Mengni Li","doi":"10.3934/cpaa.2023015","DOIUrl":"https://doi.org/10.3934/cpaa.2023015","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":"70220986","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":"Existence of at most two limit cycles for some non-autonomous differential equations","authors":"A. Gasull, Yulin Zhao","doi":"10.3934/cpaa.2023016","DOIUrl":"https://doi.org/10.3934/cpaa.2023016","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":"70221022","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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