{"title":"Multi-peak solutions for logarithmic Schrödinger equations with potentials unbounded below","authors":"Xiaoming An, Xian-lin Yang","doi":"10.3934/dcds.2023073","DOIUrl":"https://doi.org/10.3934/dcds.2023073","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77522009","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":"Modified scattering for the nonlinear nonlocal Schrödinger equation in two space dimensions","authors":"N. Hayashi, P. Naumkin","doi":"10.3934/dcds.2023065","DOIUrl":"https://doi.org/10.3934/dcds.2023065","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"54 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79264726","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":"Bistable pulsating wave of a competition model in rapidly varying media and its homogenization limit","authors":"Weiwei Ding, Rui Huang, Xiao Yu","doi":"10.3934/dcds.2023012","DOIUrl":"https://doi.org/10.3934/dcds.2023012","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"12 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77959575","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":"Vacuum and singularity formation for compressible Euler equations with time-dependent damping","authors":"Ying Sui, Weiqiang Wang, Huimin Yu","doi":"10.3934/dcds.2022184","DOIUrl":"https://doi.org/10.3934/dcds.2022184","url":null,"abstract":"In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $ 1<gamma{leq} 3 $, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $ lambda $, which was open in [19] for some cases. Moreover, the singularity formation of the compressible Euler equations when $ gamma = 3 $ is investigated, too.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"16 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":"134996609","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":"Refined spike profiles of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials","authors":"Yujin Guo, Wenning Liang, Yan Li","doi":"10.3934/dcds.2023100","DOIUrl":"https://doi.org/10.3934/dcds.2023100","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"131 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74986822","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 dispersal Leslie-Gower predator-prey model with some shifting habitats","authors":"Qinhe Fang, Hongmei Cheng, Rong Yuan","doi":"10.3934/dcds.2023037","DOIUrl":"https://doi.org/10.3934/dcds.2023037","url":null,"abstract":"","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"24 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76720267","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 are concerned with the quasi-periodic semilinear Duffing equation $ x''+omega^2x+g(x,t) = 0, $ where $ omega $ is a Diophantine number, $ g(x,t) $ is bounded, real analytic in $ x $ and $ t $, and is quasi-periodic in $ t $ with the frequency $ tilde{omega} = (1, alpha) $, where $ alpha $ is Liouvillean. Without assuming the twist condition and the polynomial-like condition on this equation, we will prove the boundedness of all solutions.
我们关注拟周期半线性Duffing方程$ x''+omega^2x+g(x,t) = 0, $，其中$ omega $是丢芬图数，$ g(x,t) $是有界的，在$ x $和$ t $是实解析的，在$ t $是拟周期的，频率为$ tilde{omega} = (1, alpha) $，其中$ alpha $是Liouvillean。在不假设该方程的扭转条件和类多项式条件的情况下，证明了所有解的有界性。
{"title":"Boundedness of semilinear Duffing equations with Liouvillean frequency","authors":"Min Li, Xiong Li","doi":"10.3934/dcds.2023127","DOIUrl":"https://doi.org/10.3934/dcds.2023127","url":null,"abstract":"We are concerned with the quasi-periodic semilinear Duffing equation $ x''+omega^2x+g(x,t) = 0, $ where $ omega $ is a Diophantine number, $ g(x,t) $ is bounded, real analytic in $ x $ and $ t $, and is quasi-periodic in $ t $ with the frequency $ tilde{omega} = (1, alpha) $, where $ alpha $ is Liouvillean. Without assuming the twist condition and the polynomial-like condition on this equation, we will prove the boundedness of all solutions.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"51 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":"135319485","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 prove a Liouville theorem for the Chern–Simons–Schrödinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [10] for the soliton resolution result in a weighted space.
{"title":"A Liouville theorem for the Chern–Simons–Schrödinger equation","authors":"Benjamin Dodson","doi":"10.3934/dcds.2023110","DOIUrl":"https://doi.org/10.3934/dcds.2023110","url":null,"abstract":"In this paper we prove a Liouville theorem for the Chern–Simons–Schrödinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [10] for the soliton resolution result in a weighted space.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"41 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":"135839031","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}
Palmer's linearization theorem states that a hyperbolic linear system is topologically conjugated to its bounded perturbation. Recently, Huerta (DCDS 2020 [8]), Castañeda and Robledo (DCDS 2018 [3]) and Lin (NA 2007 [13]) generalized Palmer's theorem to the linearization with unbounded perturbation (continuous or discrete) by assuming that the linear part of the system is contractive or nonuniformly contractive. However, these previous works sacrifice the hyperbolicity of the linear part. Is it possible to study the linearization with unbounded perturbations in the hyperbolic case? In this paper, we improve the previous works [3,8,13] to the hyperbolic unbounded systems. For the contraction, each trajectory crosses its respective unit sphere exactly once. However, for the hyperbolic system, either the trajectory does not cross the unit sphere, or the trajectory cross it twice. Thus, the standard method used in the previous works for the contractive case is not valid for the hyperbolic case yet. We develop a method to overcome the difficulty based on two 'cylinders'. Furthermore, quantitative results for the parameters are provided.
帕尔默线性化定理指出一个双曲线性系统是拓扑共轭于它的有界摄动的。最近，Huerta (DCDS 2020 [8])， Castañeda和Robledo (DCDS 2018[3])和Lin (NA 2007[13])通过假设系统的线性部分是收缩或非均匀收缩，将Palmer定理推广到具有无界摄动(连续或离散)的线性化。然而，这些先前的作品牺牲了线性部分的双曲性。有没有可能研究双曲情况下无界扰动的线性化?本文将前人的研究成果[3,8,13]改进到双曲无界系统。对于收缩，每条轨迹正好穿过它各自的单位球一次。然而，对于双曲系统，要么轨迹没有穿过单位球，要么轨迹两次穿过单位球。因此，在前面的工作中使用的标准方法的收缩情况是无效的双曲情况。我们开发了一种基于两个“圆柱体”的方法来克服困难。此外，还给出了参数的定量结果。
{"title":"Linearization of a nonautonomous unbounded system with hyperbolic linear part: A spectral approach","authors":"Mengda Wu, Yonghui Xia","doi":"10.3934/dcds.2023112","DOIUrl":"https://doi.org/10.3934/dcds.2023112","url":null,"abstract":"Palmer's linearization theorem states that a hyperbolic linear system is topologically conjugated to its bounded perturbation. Recently, Huerta (DCDS 2020 [8]), Castañeda and Robledo (DCDS 2018 [3]) and Lin (NA 2007 [13]) generalized Palmer's theorem to the linearization with unbounded perturbation (continuous or discrete) by assuming that the linear part of the system is contractive or nonuniformly contractive. However, these previous works sacrifice the hyperbolicity of the linear part. Is it possible to study the linearization with unbounded perturbations in the hyperbolic case? In this paper, we improve the previous works [3,8,13] to the hyperbolic unbounded systems. For the contraction, each trajectory crosses its respective unit sphere exactly once. However, for the hyperbolic system, either the trajectory does not cross the unit sphere, or the trajectory cross it twice. Thus, the standard method used in the previous works for the contractive case is not valid for the hyperbolic case yet. We develop a method to overcome the difficulty based on two 'cylinders'. Furthermore, quantitative results for the parameters are provided.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"68 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":"136256735","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 a coupled PDE system describing the dynamics of morphogen transport in epithelia, where the morphogens sense the spatial gradient of the logarithm of the signal following the empirically well-tested Webner–Fecher law. We prove that this fully parabolic system is globally well-posed and its unique solution is classical and uniformly bounded in time. Moreover, we find that regardless of the strength of the chemotactic motion and the size of the initial data, a linear degradation is strong enough to overcome the logarithmic singularity and destabilize the system globally and exponentially in time. Several numerical simulations are presented to illustrate and support the theoretical results.
{"title":"Global and exponential stabilization of morphogenesis models with logarithmic sensitivity and linear degradation","authors":"Lin Chen, Fanze Kong, Qi Wang","doi":"10.3934/dcds.2023115","DOIUrl":"https://doi.org/10.3934/dcds.2023115","url":null,"abstract":"We study a coupled PDE system describing the dynamics of morphogen transport in epithelia, where the morphogens sense the spatial gradient of the logarithm of the signal following the empirically well-tested Webner–Fecher law. We prove that this fully parabolic system is globally well-posed and its unique solution is classical and uniformly bounded in time. Moreover, we find that regardless of the strength of the chemotactic motion and the size of the initial data, a linear degradation is strong enough to overcome the logarithmic singularity and destabilize the system globally and exponentially in time. Several numerical simulations are presented to illustrate and support the theoretical results.","PeriodicalId":51007,"journal":{"name":"Discrete and Continuous Dynamical Systems","volume":"50 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":"136366912","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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