{"title":"Liouville type theorems for anisotropic degenerate elliptic equations on strips","authors":"Luisa Moschini","doi":"10.3934/cpaa.2023083","DOIUrl":"https://doi.org/10.3934/cpaa.2023083","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":"70221471","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 solutions for coupled hybrid systems of differential equations for microscopic dynamics and local concentrations","authors":"M. Menci, M. Papi, Flavia Smarrazzo","doi":"10.3934/cpaa.2023061","DOIUrl":"https://doi.org/10.3934/cpaa.2023061","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":"70221564","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":"Invariant tori for the Hamiltonian derivative wave equation with higher order nonlinearity","authors":"M. Gao","doi":"10.3934/cpaa.2023033","DOIUrl":"https://doi.org/10.3934/cpaa.2023033","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":"70221602","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":"Liouville theorems for Hénon type Choquard Equation","authors":"Jing Dong, Haiyang He","doi":"10.3934/cpaa.2023101","DOIUrl":"https://doi.org/10.3934/cpaa.2023101","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"17 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":"135317657","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 paper, we consider the weighted integral system with negative exponents on the upper half space $ mathbb{R}^{n+1}_+ $ as follows$ begin{equation*} begin{cases} u(X) = displaystyle{int}_{mathbb{R}^{n+1}_+}frac{f(u, v)(Y)}{t^alpha z^beta|X-Y|^lambda}dY, &Xinmathbb{R}^{n+1}_+, v(X) = displaystyle{int}_{mathbb{R}^{n+1}_+}frac{g(u, v)(Y)}{ t^beta z^alpha|X-Y|^lambda}dY, &Xinmathbb{R}^{n+1}_+, end{cases} end{equation*} $where $ alpha, betale0 $, $ lambda<0 $ and $ X = (x, t), , Y = (y, z). $ Under the natural conditions on $ f $ and $ g $, we obtain the classification and symmetry of positive solutions by the method of moving spheres in integral forms. Moreover, we generalize our results to integral system on $ mathbb{R}^{n+m} $.
{"title":"Liouville type theorems for general weighted integral system with negative exponents","authors":"Jingjing Ma, Yunyun Hu","doi":"10.3934/cpaa.2023103","DOIUrl":"https://doi.org/10.3934/cpaa.2023103","url":null,"abstract":"In this paper, we consider the weighted integral system with negative exponents on the upper half space $ mathbb{R}^{n+1}_+ $ as follows$ begin{equation*} begin{cases} u(X) = displaystyle{int}_{mathbb{R}^{n+1}_+}frac{f(u, v)(Y)}{t^alpha z^beta|X-Y|^lambda}dY, &Xinmathbb{R}^{n+1}_+, v(X) = displaystyle{int}_{mathbb{R}^{n+1}_+}frac{g(u, v)(Y)}{ t^beta z^alpha|X-Y|^lambda}dY, &Xinmathbb{R}^{n+1}_+, end{cases} end{equation*} $where $ alpha, betale0 $, $ lambda<0 $ and $ X = (x, t), , Y = (y, z). $ Under the natural conditions on $ f $ and $ g $, we obtain the classification and symmetry of positive solutions by the method of moving spheres in integral forms. Moreover, we generalize our results to integral system on $ mathbb{R}^{n+m} $.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"43 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":"135496199","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 investigate the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9,11,21].
{"title":"Existence and nonexistence of global solutions for time-dependent damped NLS equations","authors":"Makram Hamouda, Mohamed Majdoub","doi":"10.3934/cpaa.2023100","DOIUrl":"https://doi.org/10.3934/cpaa.2023100","url":null,"abstract":"We investigate the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9,11,21].","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"66 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":"135496191","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 the note, the Euler scaling is used to study a certain scenario of potential Type Ⅱ blowups of solutions to the Navier-Stokes equations.
在笔记中，欧拉标度用于研究Navier-Stokes方程解的潜在Ⅱ型爆炸的特定场景。
{"title":"Remarks on Type Ⅱ blowups of solutions to the Navier-Stokes equations","authors":"Gregory Seregin","doi":"10.3934/cpaa.2023108","DOIUrl":"https://doi.org/10.3934/cpaa.2023108","url":null,"abstract":"In the note, the Euler scaling is used to study a certain scenario of potential Type Ⅱ blowups of solutions to the Navier-Stokes equations.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"300 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":"135594478","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 a complexification of the Euler equations introduced by Šverák in [35] which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions which lose analyticity in finite time. Our examples are complex shear flows and, hence, one-dimensional. This motivates us to consider fully nonlinear systems in one spatial dimension which are non-hyperbolic near a constant equilibrium. We prove nonlinear ill-posedness and finite-time singularity for these models. Our approach is to construct an infinite-dimensional unstable manifold to capture the high frequency instability at the nonlinear level.
{"title":"Remarks on the complex Euler equations","authors":"Dallas Albritton, W. Jacob Ogden","doi":"10.3934/cpaa.2023110","DOIUrl":"https://doi.org/10.3934/cpaa.2023110","url":null,"abstract":"We consider a complexification of the Euler equations introduced by Šverák in [35] which conserves energy. We prove that these complex Euler equations are nonlinearly ill-posed below analytic regularity and, moreover, we exhibit solutions which lose analyticity in finite time. Our examples are complex shear flows and, hence, one-dimensional. This motivates us to consider fully nonlinear systems in one spatial dimension which are non-hyperbolic near a constant equilibrium. We prove nonlinear ill-posedness and finite-time singularity for these models. Our approach is to construct an infinite-dimensional unstable manifold to capture the high frequency instability at the nonlinear level.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"45 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":"136053986","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":"Multiple nonsemitrivial solutions for quasilinear elliptic systems via associated eigenvalue problems","authors":"Ying-Chieh Lin, Kuan‐Hsiang Wang, Tsung‐fang Wu","doi":"10.3934/cpaa.2023041","DOIUrl":"https://doi.org/10.3934/cpaa.2023041","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":"70221283","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":"The weighted Yamabe flow with boundary","authors":"P. Ho, Jin‐Hyuk Shin, Zetian Yan","doi":"10.3934/cpaa.2023079","DOIUrl":"https://doi.org/10.3934/cpaa.2023079","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":"70221314","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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