Pub Date : 2022-09-11DOI: 10.4310/dpde.2023.v20.n3.a3
S. Gustafson, Takahisa Inui
We consider the nonlinear Schr"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.
{"title":"Blow-up or Grow-up for the threshold solutions to the nonlinear Schrödinger equation","authors":"S. Gustafson, Takahisa Inui","doi":"10.4310/dpde.2023.v20.n3.a3","DOIUrl":"https://doi.org/10.4310/dpde.2023.v20.n3.a3","url":null,"abstract":"We consider the nonlinear Schr\"{o}dinger equation with $L^{2}$-supercritical and $H^{1}$-subcritical power type nonlinearity. Duyckaerts and Roudenko and Campos, Farah, and Roudenko studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48155812","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}
Pub Date : 2022-07-01DOI: 10.4310/dpde.2023.v20.n2.a4
Y. Sire, Xueying Yu, H. Yue, Zehua Zhao
We introduce new models for Schr"odinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $mathbb{R}^d times mathbb{T}$, $d geq 3$. This result can be regarded as the analogue of cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.
{"title":"Singular Levy processes and dispersive effects of generalized Schrödinger equations","authors":"Y. Sire, Xueying Yu, H. Yue, Zehua Zhao","doi":"10.4310/dpde.2023.v20.n2.a4","DOIUrl":"https://doi.org/10.4310/dpde.2023.v20.n2.a4","url":null,"abstract":"We introduce new models for Schr\"odinger-type equations, which generalize standard NLS and for which different dispersion occurs depending on the directions. Our purpose is to understand dispersive properties depending on the directions of propagation, in the spirit of waveguide manifolds, but where the diffusion is of different types. We mainly consider the standard Euclidean space and the waveguide case but our arguments extend easily to other types of manifolds (like product spaces). Our approach unifies in a natural way several previous results. Those models are also generalizations of some appearing in seminal works in mathematical physics, such as relativistic strings. In particular, we prove the large data scattering on waveguide manifolds $mathbb{R}^d times mathbb{T}$, $d geq 3$. This result can be regarded as the analogue of cite{TV2, YYZ2} in our setting and the waveguide analogue investigated in cite{GSWZ}. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46737887","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}
Pub Date : 2022-06-26DOI: 10.4310/DPDE.2023.v20.n1.a4
A. Bolsinov, A. Konyaev, V. Matveev
We construct a new series of multicomponent integrable PDE systems that con-tain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV [1], Harry Dym, coupled Harry Dym [2], Camassa-Holm, multicomponent Camassa-Holm [14], Kaup-Boussinesq systems. The series contains also integrable systems with no low-component analogues.
{"title":"Applications of Nijenhuis geometry IV: Multicomponent KdV and Camassa–Holm equations","authors":"A. Bolsinov, A. Konyaev, V. Matveev","doi":"10.4310/DPDE.2023.v20.n1.a4","DOIUrl":"https://doi.org/10.4310/DPDE.2023.v20.n1.a4","url":null,"abstract":"We construct a new series of multicomponent integrable PDE systems that con-tain as particular example (with appropriately chosen parameters) many famous integrable systems including KdV, coupled KdV [1], Harry Dym, coupled Harry Dym [2], Camassa-Holm, multicomponent Camassa-Holm [14], Kaup-Boussinesq systems. The series contains also integrable systems with no low-component analogues.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44613830","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}
Pub Date : 2022-03-20DOI: 10.4310/DPDE.2023.v20.n3.a1
Igor Leite Freire
Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions of the system in a suitable Banach space. The second one is regarded to well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the $b-$; Fornberg-Whitham; potential and $pi-$Camassa-Holm; generalised Boussinesq equations; and the modified Euler-Poisson system.
{"title":"Unique continuation results for abstract quasi-linear evolution equations in Banach spaces","authors":"Igor Leite Freire","doi":"10.4310/DPDE.2023.v20.n3.a1","DOIUrl":"https://doi.org/10.4310/DPDE.2023.v20.n3.a1","url":null,"abstract":"Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions of the system in a suitable Banach space. The second one is regarded to well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the $b-$; Fornberg-Whitham; potential and $pi-$Camassa-Holm; generalised Boussinesq equations; and the modified Euler-Poisson system.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45799311","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n3.a1
Anna Michalak, A. Nowakowski
{"title":"Dual Lyapunov approach to finite time stability for parabolic PDE","authors":"Anna Michalak, A. Nowakowski","doi":"10.4310/dpde.2022.v19.n3.a1","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n3.a1","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427216","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n4.a1
Tingting Ma, Xinzhu Meng
{"title":"Dynamic analysis of a diffusive eco-epidemiological system with fear effect and prey refuge","authors":"Tingting Ma, Xinzhu Meng","doi":"10.4310/dpde.2022.v19.n4.a1","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n4.a1","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427264","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n4.a5
Jingbo Wu, Xiaoqin Nan, Yan Jia
{"title":"Global regularity of magneto-micropolar equations with logarithmically dissipation","authors":"Jingbo Wu, Xiaoqin Nan, Yan Jia","doi":"10.4310/dpde.2022.v19.n4.a5","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n4.a5","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70426912","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n3.a4
Yeping Li, Yujie Qian, Shengqi Yu
. In this article, we are concerned with the large-time behavior of solutions to an inflow problem in one-dimensional case for the Navier-Stokes- Korteweg equation, which models compressible fluids with internal capillarity. We first investigate that the asymptotic state is the rarefaction wave under the proper condition of the far fields and boundary values. The asymptotic stability of the rarefaction wave under some smallness conditions is shown. The proof is completed by the energy method with the help of time-decay estimate for the rarefaction wave.
{"title":"Asymptotics toward rarefaction wave for an inflow problem of the compressible Navier–Stokes–Korteweg equation","authors":"Yeping Li, Yujie Qian, Shengqi Yu","doi":"10.4310/dpde.2022.v19.n3.a4","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n3.a4","url":null,"abstract":". In this article, we are concerned with the large-time behavior of solutions to an inflow problem in one-dimensional case for the Navier-Stokes- Korteweg equation, which models compressible fluids with internal capillarity. We first investigate that the asymptotic state is the rarefaction wave under the proper condition of the far fields and boundary values. The asymptotic stability of the rarefaction wave under some smallness conditions is shown. The proof is completed by the energy method with the help of time-decay estimate for the rarefaction wave.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427252","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n3.a3
P. Han, Keke Lei, Chenggang Liu, Xuewen Wang
. In this paper, we study the asymptotic behavior of short trajectories of weak solutions to the 2D nonhomogeneous heat-conducting magneto- hydrodynamic equations. Several bounds for short trajectories are obtained. An attracting set is constructed, which consists of orbits on [0 , 1] of complete bounded solutions. Furthermore, the attracting set is compact in different topologies.
{"title":"Asymptotic behavior of short trajectories to nonhomogeneous heat-conducting magnetohydrodynamic equations","authors":"P. Han, Keke Lei, Chenggang Liu, Xuewen Wang","doi":"10.4310/dpde.2022.v19.n3.a3","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n3.a3","url":null,"abstract":". In this paper, we study the asymptotic behavior of short trajectories of weak solutions to the 2D nonhomogeneous heat-conducting magneto- hydrodynamic equations. Several bounds for short trajectories are obtained. An attracting set is constructed, which consists of orbits on [0 , 1] of complete bounded solutions. Furthermore, the attracting set is compact in different topologies.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427236","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}
Pub Date : 2022-01-01DOI: 10.4310/dpde.2022.v19.n4.a3
Miguel A. Fontecha-Medina, É. J. Villamizar-Roa
{"title":"Global existence and asymptotic behavior of solutions for a fractional chemotaxis-Navier-Stokes system","authors":"Miguel A. Fontecha-Medina, É. J. Villamizar-Roa","doi":"10.4310/dpde.2022.v19.n4.a3","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n4.a3","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70427348","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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