Global dynamics below the ground state for the focusing Schr\"odinger equation with a potential
Global dynamics below the ground state for the focusing Schr\"odinger equation with a potential
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具有势能的聚焦薛定格方程的基态以下的全局动力学
DOI：
10.1007/s00028-019-00547-z
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发表时间：
2019
影响因子：
1.4
通讯作者：
Masahiro Ikeda
中科院分区：
文献类型：
--
作者：
Masaru Hamano;Masahiro Ikeda
本文研究了具有实值位势的非线性薛定谔方程。在势V满足一定条件的情况下，我们研究了方程在基态以下解的整体行为，证明了在质量超临界和能量亚临界下的一个散射结果和一个爆破结果.我们对散射结果的证明是基于Dodson-Murphy的论点（Proc Am Math Soc 145（11）：4859-4867，2017）。在没有径向对称假设的情况下，爆破或增长结果的证明基于Du-Wu-Zhang的论点（Discrete Contin Dyn Syst 36（7）：3639-3650，2016）。我们可以通过论证排除增长结果的可能性（Inui等人，在Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrodinger system without mass resonance，arXiv：1810.09153.; Nakanishi和Schlag在Calc Var Partial Differ Eq 44（1-2）：1-45，2012; Alfressey在J Math Phys 18（9）：1794-1797，1977中），如果“数据和势是径向对称的”或“数据具有有限方差”。
In this paper, we consider the nonlinear Schrödinger equation with a real-valued potential. We study global behavior of solutions to the equation with data below the ground state under some conditions for the potentialVand prove a scattering result and a blowing-up result in mass-supercritical and energy-subcritical. Our proof of the scattering result is based on an argument by Dodson–Murphy (Proc Am Math Soc 145(11):4859–4867, 2017). The proof of the blowing-up or growing-up result without radially symmetric assumption is based on the argument by Du–Wu–Zhang (Discrete Contin Dyn Syst 36(7):3639–3650, 2016). We can exclude the possibility of the growing-up result by the argument (Inui et al. in Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrodinger system without mass resonance, arXiv:1810.09153.; Nakanishi and Schlag in Calc Var Partial Differ Eq 44(1-2):1–45, 2012; Glassey in J Math Phys 18(9):1794–1797, 1977) if “the data and the potential are radially symmetric” or “the data has finite variance.”