Pub Date : 2020-12-24DOI: 10.4310/dpde.2022.v19.n1.a1
O. Landoulsi
In this paper, we consider the Schrödinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of R with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on R. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obstacle because of the appearance of boundary terms with an unfavorable sign in the second derivative of the variance. The main idea of our proof is to introduce a new modified variance which is bounded from below and strictly concave for the solutions that we consider.
{"title":"On blow-up solutions to the nonlinear Schrödinger equation in the exterior of a convex obstacle","authors":"O. Landoulsi","doi":"10.4310/dpde.2022.v19.n1.a1","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n1.a1","url":null,"abstract":"In this paper, we consider the Schrödinger equation with a mass-supercritical focusing nonlinearity, in the exterior of a smooth, compact, convex obstacle of R with Dirichlet boundary conditions. We prove that solutions with negative energy blow up in finite time. Assuming furthermore that the nonlinearity is energy-subcritical, we also prove (under additional symmetry conditions) blow-up with the same optimal ground-state criterion than in the work of Holmer and Roudenko on R. The classical proof of Glassey, based on the concavity of the variance, fails in the exterior of an obstacle because of the appearance of boundary terms with an unfavorable sign in the second derivative of the variance. The main idea of our proof is to introduce a new modified variance which is bounded from below and strictly concave for the solutions that we consider.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44488636","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":"Second- and Higher-Order Linear Partial Differential Equations","authors":"Nita H. Shah, Mrudul Y. Jani","doi":"10.1201/9781003105183-3","DOIUrl":"https://doi.org/10.1201/9781003105183-3","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86561966","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":"Introduction of Partial Differential Equations","authors":"Nita H. Shah, Mrudul Y. Jani","doi":"10.1201/9781003105183-1","DOIUrl":"https://doi.org/10.1201/9781003105183-1","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77560264","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":"Applications of Partial Differential Equations","authors":"Nita H. Shah, Mrudul Y. Jani","doi":"10.1201/9781003105183-4","DOIUrl":"https://doi.org/10.1201/9781003105183-4","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83744295","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 : 2020-08-01DOI: 10.4310/dpde.2022.v19.n1.a4
F. Cacciafesta, Zhiqing Yin, Junyong Zhang
We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.
{"title":"Generalized Strichartz estimates for wave and Dirac equations in Aharonov–Bohm magnetic fields","authors":"F. Cacciafesta, Zhiqing Yin, Junyong Zhang","doi":"10.4310/dpde.2022.v19.n1.a4","DOIUrl":"https://doi.org/10.4310/dpde.2022.v19.n1.a4","url":null,"abstract":"We prove generalized Strichartz estimates for wave and massless Dirac equations in Aharonov-Bohm magnetic fields. Following a well established strategy to deal with scaling critical perturbations of dispersive PDEs, we make use of Hankel transform and rely on some precise estimates on Bessel functions. As a complementary result, we prove a local smoothing estimate for the Klein-Gordon equation in the same magnetic field.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47405270","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 : 2020-06-23DOI: 10.4310/DPDE.2021.V18.N2.A2
A. Boukarou, Daniel Oliveira da Silva, K. Guerbati, K. Zennir
We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.
{"title":"Global well-posedness for the fifth-order Kadomtsev–Petviashvili II equation in anisotropic Gevrey spaces","authors":"A. Boukarou, Daniel Oliveira da Silva, K. Guerbati, K. Zennir","doi":"10.4310/DPDE.2021.V18.N2.A2","DOIUrl":"https://doi.org/10.4310/DPDE.2021.V18.N2.A2","url":null,"abstract":"We show that the fifth-order Kadomtsev-Petviashvili II equation is globally well-posed in an anisotropic Gevrey space, which complements earlier results on the well-posedness of this equation in anisotropic Sobolev spaces.","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45025870","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 : 2020-06-08DOI: 10.1515/9783110677256-007
{"title":"7. Method of characteristics for first-order quasilinear equations","authors":"","doi":"10.1515/9783110677256-007","DOIUrl":"https://doi.org/10.1515/9783110677256-007","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88236942","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 : 2020-06-08DOI: 10.1515/9783110677256-003
{"title":"3. First-order linear equations","authors":"","doi":"10.1515/9783110677256-003","DOIUrl":"https://doi.org/10.1515/9783110677256-003","url":null,"abstract":"","PeriodicalId":50562,"journal":{"name":"Dynamics of Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79026425","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：