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}
{"title":"Hopf bifurcation in a two-species reaction-diffusion-advection competitive model with nonlocal delay","authors":"Tingting Wen, Xiaoli Wang, Guohong Zhang","doi":"10.3934/cpaa.2023036","DOIUrl":"https://doi.org/10.3934/cpaa.2023036","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":"70221177","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":"Infinitely many large energy solutions to a class of nonlocal discrete elliptic boundary value problems","authors":"Y. Long, Qinqin Zhang","doi":"10.3934/cpaa.2023037","DOIUrl":"https://doi.org/10.3934/cpaa.2023037","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":"70221217","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":"A remark on the local uniqueness by the monotonicity based method for the magnetic Schrödinger equation","authors":"Ryusei Yamashita","doi":"10.3934/cpaa.2023077","DOIUrl":"https://doi.org/10.3934/cpaa.2023077","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":"70221303","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":"Symmetry and monotonicity of positive solutions to systems involving general pseudo-relativistic operators","authors":"Xueying Chen, Guanfeng Li","doi":"10.3934/cpaa.2023045","DOIUrl":"https://doi.org/10.3934/cpaa.2023045","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":"70221345","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 positive solutions to critical Schrödinger system with mixed interactions in $ {mathbb R}^3 $","authors":"Song You, Houwang Li","doi":"10.3934/cpaa.2023097","DOIUrl":"https://doi.org/10.3934/cpaa.2023097","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"478 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221766","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 study the emergent discrete-time dynamics of the Kuramoto model with frustration. For the discrete identical Kuramoto model with frustration, we present the exponential synchronization estimate for some admissible classes of initial configuration confined in a half circle. For the nonidentical oscillators, we exploit the second-order lifting of the original Kuramoto model with frustration and study the emergent behaviors of the corresponding discrete augmented model. More precisely, under a sufficient framework in terms of large coupling strength and small step size, we derive that the initial configuration distributed in a small set of a quarter circle will converge to frequency synchronization.
{"title":"On the synchronization of discrete-time Kuramoto model with frustration","authors":"Tingting Zhu","doi":"10.3934/cpaa.2023109","DOIUrl":"https://doi.org/10.3934/cpaa.2023109","url":null,"abstract":"In this paper, we study the emergent discrete-time dynamics of the Kuramoto model with frustration. For the discrete identical Kuramoto model with frustration, we present the exponential synchronization estimate for some admissible classes of initial configuration confined in a half circle. For the nonidentical oscillators, we exploit the second-order lifting of the original Kuramoto model with frustration and study the emergent behaviors of the corresponding discrete augmented model. More precisely, under a sufficient framework in terms of large coupling strength and small step size, we derive that the initial configuration distributed in a small set of a quarter circle will converge to frequency synchronization.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"36 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":"135700499","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}
Lyapunov exponent plays an important role in the dynamics of quasi-periodic cocycles, and admits an asymptotic formula for Schrödinger cocycle with large parameter (see [12] for analytic case, and see [15] for finitely smooth case). In this paper, we establish similar asymptotic formulas of Lyapunov exponent for two classes of important quasi-periodic cocycles: Szegő cocycles and Schrödinger Mosaic cocycles.
{"title":"On the Lyapunov exponent for some quasi-periodic cocycles with large parameter","authors":"Bowen Ding, Jinhao Liang","doi":"10.3934/cpaa.2023102","DOIUrl":"https://doi.org/10.3934/cpaa.2023102","url":null,"abstract":"Lyapunov exponent plays an important role in the dynamics of quasi-periodic cocycles, and admits an asymptotic formula for Schrödinger cocycle with large parameter (see [12] for analytic case, and see [15] for finitely smooth case). In this paper, we establish similar asymptotic formulas of Lyapunov exponent for two classes of important quasi-periodic cocycles: Szegő cocycles and Schrödinger Mosaic cocycles.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"254 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":"135496200","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":"Global dynamics of a two-species competition model in advective homogeneous environments","authors":"Jinyu Wei, Bin Liu","doi":"10.3934/cpaa.2023021","DOIUrl":"https://doi.org/10.3934/cpaa.2023021","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":"70221118","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":"Spatial Dynamics of a Nonlocal Reaction-Diffusion Epidemic Model in Time-Space Periodic Habitat","authors":"Ming-Zhen Xin, Bin-Guo Wang","doi":"10.3934/cpaa.2023072","DOIUrl":"https://doi.org/10.3934/cpaa.2023072","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"36 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70221252","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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