{"title":"Global existence and Scattering for nonlinear Schrödinger equations with time-dependent damping","authors":"Chourouk Bamri, S. Tayachi","doi":"10.3934/cpaa.2023069","DOIUrl":"https://doi.org/10.3934/cpaa.2023069","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":"70221195","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 velocity alignment model with a bonding force on complete Riemannian manifolds","authors":"Zhengyang Qiao, Yicheng Liu, Xiao Wang","doi":"10.3934/cpaa.2023091","DOIUrl":"https://doi.org/10.3934/cpaa.2023091","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":"70221682","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":"Erratum to \"The combined effect in one space dimension beyond the general theory for nonlinear wave equations\"","authors":"K. Morisawa, Takiko Sasaki, H. Takamura","doi":"10.3934/cpaa.2023096","DOIUrl":"https://doi.org/10.3934/cpaa.2023096","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":"70221755","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":"Periodic solutions to a nonlinear Dirac-Klein-Gordon system with concave and convex nonlinearities","authors":"Hua-Yang Wang, Yanheng Ding, Pan Chen","doi":"10.3934/cpaa.2023046","DOIUrl":"https://doi.org/10.3934/cpaa.2023046","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":"70221384","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}
Duván Cardona, Julio Delgado, Brian Grajales, Michael Ruzhansky
Let $ A $ and $ B $ be invariant linear operators with respect to a decomposition $ {H_{j}}_{jin mathbb{N}} $ of a Hilbert space $ mathcal{H} $ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem$ u_t = Au+Bv, , , u(0) = u_0, $in terms of the (global) matrix-valued symbols $ sigma_A $ and $ sigma_B $ of $ A $ and $ B, $ respectively, associated to the decomposition $ {H_{j}}_{jin mathbb{N}} $. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllability of wave and Schrödinger equations in these settings.
设$ A $和$ B $是对有限维子空间中希尔伯特空间$ mathcal{H} $的分解$ {H_{j}}_{j mathbb{N} $的不变线性算子。本文给出了柯西问题$ u_t = Au+Bv， ， ， u(0) = u_0， $在$ A $和$ B $的(全局)矩阵值符号$ sigma_A $和$ sigma_B $的可控性的充分必要条件，这些可控性与$ mathbb{N}} $中的分解$ {H_{j}}_{j有关。然后，我们给出了紧流形上Cauchy问题对椭圆算子的可控性和紧李群上Hörmander子拉普拉斯的分数扩散模型的可控性的一些应用。我们也给出了波的可控性的条件和Schrödinger方程。
{"title":"Control of the Cauchy problem on Hilbert spaces: A global approach via symbol criteria","authors":"Duván Cardona, Julio Delgado, Brian Grajales, Michael Ruzhansky","doi":"10.3934/cpaa.2023113","DOIUrl":"https://doi.org/10.3934/cpaa.2023113","url":null,"abstract":"Let $ A $ and $ B $ be invariant linear operators with respect to a decomposition $ {H_{j}}_{jin mathbb{N}} $ of a Hilbert space $ mathcal{H} $ in subspaces of finite dimension. We give necessary and sufficient conditions for the controllability of the Cauchy problem$ u_t = Au+Bv, , , u(0) = u_0, $in terms of the (global) matrix-valued symbols $ sigma_A $ and $ sigma_B $ of $ A $ and $ B, $ respectively, associated to the decomposition $ {H_{j}}_{jin mathbb{N}} $. Then, we present some applications including the controllability of the Cauchy problem on compact manifolds for elliptic operators and the controllability of fractional diffusion models for Hörmander sub-Laplacians on compact Lie groups. We also give conditions for the controllability of wave and Schrödinger equations in these settings.","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":"135057391","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":"Dynamics of an evolution equation with singular potential","authors":"Renato Colucci","doi":"10.3934/cpaa.2023117","DOIUrl":"https://doi.org/10.3934/cpaa.2023117","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"122 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":"135317637","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 the energy-critical nonlinear Schrödinger equation with randomised initial data in dimensions $ d>6 $. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $ H^s(mathbb{R}^d) $ whenever $ s>max{frac{4d-1}{3(2d-1)},frac{d^2+6d-4}{(2d-1)(d+2)}} $. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results in dimension 4 [18]. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.
{"title":"Almost sure scattering of the energy-critical NLS in $ d>6 $","authors":"Katie Marsden","doi":"10.3934/cpaa.2023106","DOIUrl":"https://doi.org/10.3934/cpaa.2023106","url":null,"abstract":"We study the energy-critical nonlinear Schrödinger equation with randomised initial data in dimensions $ d>6 $. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $ H^s(mathbb{R}^d) $ whenever $ s>max{frac{4d-1}{3(2d-1)},frac{d^2+6d-4}{(2d-1)(d+2)}} $. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results in dimension 4 [18]. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"40 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":"135549390","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 radial solutions of $ Delta u + K(|x|) f(u) = 0 $ in the exterior of the ball of radius $ R>0 $ in $ {mathbb R}^{N} $ where $ f $ grows superlinearly at infinity and is singular at $ 0 $ with $ f(u) sim -frac{1}{|u|^{q-1}u} $ and $ 0
{"title":"Existence and nonexistence of sign-changing solutions for singular superlinear equations on exterior domains","authors":"Joseph Iaia","doi":"10.3934/cpaa.2023107","DOIUrl":"https://doi.org/10.3934/cpaa.2023107","url":null,"abstract":"In this paper we study radial solutions of $ Delta u + K(|x|) f(u) = 0 $ in the exterior of the ball of radius $ R>0 $ in $ {mathbb R}^{N} $ where $ f $ grows superlinearly at infinity and is singular at $ 0 $ with $ f(u) sim -frac{1}{|u|^{q-1}u} $ and $ 0<q<1 $ for small $ u $. We assume $ K(|x|) sim |x|^{-alpha} $ for large $ |x| $ and establish existence of an infinite number of sign-changing solutions when $ N+q(N-2) <alpha <2(N-1). $ We also prove nonexistence for $ 0<alpha < N+ q(N-2) $.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"33 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":"135550567","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}
This paper is devoted to the study of periodic solutions to a semilinear variable coefficients wave equation with resonance potential under the periodic boundary conditions. This mathematical model is used to account for the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. We characterize some properties of the spectrum of the variable coefficients wave operator for the periodic boundary conditions, especially we prove the existence of the essential spectral point. Under some suitable assumptions on the resonant potential, by combining variational methods and the Galerkin approximation argument, we get the infinitely many small periodic solutions of the resonant problem. Our results contain the case of zero potential and can be applied to the corresponding classical one dimensional wave equation.
{"title":"Existence of infinitely many small periodic solutions for a semilinear variable coefficients wave equation with resonant potential","authors":"Hui Wei","doi":"10.3934/cpaa.2023111","DOIUrl":"https://doi.org/10.3934/cpaa.2023111","url":null,"abstract":"This paper is devoted to the study of periodic solutions to a semilinear variable coefficients wave equation with resonance potential under the periodic boundary conditions. This mathematical model is used to account for the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. We characterize some properties of the spectrum of the variable coefficients wave operator for the periodic boundary conditions, especially we prove the existence of the essential spectral point. Under some suitable assumptions on the resonant potential, by combining variational methods and the Galerkin approximation argument, we get the infinitely many small periodic solutions of the resonant problem. Our results contain the case of zero potential and can be applied to the corresponding classical one dimensional wave equation.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","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":"136053981","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":"Pointwise estimates for systems of coupled $ p $-Laplacian elliptic equations","authors":"A. Aghajani, H. Shahgholian","doi":"10.3934/cpaa.2023013","DOIUrl":"https://doi.org/10.3934/cpaa.2023013","url":null,"abstract":"","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70220973","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：