{"title":"Scattering of an inhomogeneous coupled Schrödinger system in the conformal space","authors":"T. Saanouni, Congming Peng","doi":"10.1515/anly-2023-0027","DOIUrl":null,"url":null,"abstract":"Abstract This paper studies the inhomogeneous defocusing coupled Schrödinger system i u ˙ j + Δ u j = | x | - ρ ( ∑ 1 ≤ k ≤ m a j k | u k | p ) | u j | p - 2 u j , ρ > 0 , j ∈ [ 1 , m ] . i\\dot{u}_{j}+\\Delta u_{j}=\\lvert x\\rvert^{-\\rho}\\bigg{(}\\sum_{1\\leq k\\leq m}a_% {jk}\\lvert u_{k}\\rvert^{p}\\biggr{)}\\lvert u_{j}\\rvert^{p-2}u_{j},\\quad\\rho>0,% \\,j\\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ( ℝ N ) {f\\in H^{1}(\\mathbb{R}^{N})} such that x f ∈ L 2 ( ℝ N ) {xf\\in L^{2}(\\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x u 0 ∈ L 2 {xu_{0}\\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0}<p\\leq\\frac{2-\\rho}{N}+1} , where p 0 {p_{0}} is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ∥ u ( t ) ∥ L r ( ℝ N ) ≲ 1 \\lvert t\\rvert^{\\alpha}\\lVert u(t)\\rVert_{L^{r}(\\mathbb{R}^{N})}\\lesssim 1 for some α > 0 {\\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ u 0 {e^{i\\cdot\\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}<p<\\frac{2-\\rho}{N-2}+1} . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 {p\\geq 2} , which restricts the space dimensions to N ≤ 3 {N\\leq 3} . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 {\\rho<0} .","PeriodicalId":82310,"journal":{"name":"Philosophic research and analysis","volume":"47 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Philosophic research and analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/anly-2023-0027","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract This paper studies the inhomogeneous defocusing coupled Schrödinger system i u ˙ j + Δ u j = | x | - ρ ( ∑ 1 ≤ k ≤ m a j k | u k | p ) | u j | p - 2 u j , ρ > 0 , j ∈ [ 1 , m ] . i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x f ∈ L 2 ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0} 0 {\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}
摘要研究了非均匀散焦耦合Schrödinger系统i¹u˙j + Δ¹u j = | x | - ρ≠(∑1≤k≤m a j≠k≠u k | p)≠u j | p - 2∑u j， ρ > 0, j∈[1,m]。I \dot{u} ＿j{+ }\Delta ＿j{= }\lvert x \rvert ^{-\rho}\bigg{(}\sum ＿1{\leq k \leq ma}＿% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x f ∈ L 2 ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0} 0 {\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}
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