scholarly journals On Asymptotic Properties of Semi-relativistic Hartree Equation with combined Hartree-type nonlinearities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingxuan Wang ◽  
Binhua Feng ◽  
Yuan Li ◽  
Qihong Shi
Keyword(s):  
Ground State ◽  
Blow Up ◽  
Asymptotic Properties ◽  
Ground States ◽  
Threshold Value ◽  
Hartree Equation ◽  
Existence And Stability ◽  
Blow Up Rate

<p style='text-indent:20px;'>We consider the semi-relativistic Hartree equation with combined Hartree-type nonlinearities given by</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\partial_t \psi = \sqrt{-\triangle+m^2}\, \psi+\beta(\frac{1}{|x|^\alpha}\ast |\psi|^2)\psi-(\frac{1}{|x|}\ast |\psi|^2)\psi\ \ \ \text{on $\mathbb{R}^3$.} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ 0&lt;\alpha&lt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \beta&gt;0 $\end{document}</tex-math></inline-formula>. Firstly we study the existence and stability of the maximal ground state <inline-formula><tex-math id="M3">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M4">\begin{document}$ N = N_c $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ N_c $\end{document}</tex-math></inline-formula> is a threshold value and can be regarded as "Chandrasekhar limiting mass". Secondly, we analyse blow-up behaviours of maximal ground states <inline-formula><tex-math id="M6">\begin{document}$ \psi_\beta $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M7">\begin{document}$ \beta\rightarrow 0^+ $\end{document}</tex-math></inline-formula>, and the optimal blow-up rate with respect to <inline-formula><tex-math id="M8">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> will be calculated.</p>

ON ONE BLOW UP POINT SOLUTIONS TO THE CRITICAL NONLINEAR SCHRÖDINGER EQUATION

2005 ◽  
Vol 02 (04) ◽  
pp. 919-962 ◽  
Author(s):  
FRANK MERLE ◽  
PIERRE RAPHAEL
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Ground State ◽  
Finite Time ◽  
Schrodinger Equation ◽  
Blow Up ◽  
Nonlinear Schrodinger Equation ◽  
Specific Class ◽  
Nonlinear Schrödinger ◽  
Existence And Stability

We consider the L2 critical nonlinear Schrödinger equation [Formula: see text] in the energy space H1. In the series of papers [11–15,18], we studied finite time blow up solutions for which lim t↑T < + ∞ |∇ u(t)|L2 = + ∞ and proved classification results of the blow up dynamics for the specific class of small super critical L2 mass initial data. We extend these results here to a wider class of finite time blow up solutions corresponding to the ones which accumulate at one point exactly the ground state mass. In particular, we prove the existence and stability of large L2 mass log-log type solutions which are believed to describe the generic blow up dynamics.

On mass concentration for the critical generalized Korteweg–de Vries equation

2015 ◽  
Vol 59 (2) ◽  
pp. 519-532
Author(s):  
B. Pigott
Keyword(s):  
Ground State ◽  
Mass Concentration ◽  
Blow Up ◽  
Vries Equation ◽  
De Vries ◽  
Concentration Result ◽  
Blow Up Rate

AbstractWe show that blow-up solutions of the critical generalized Korteweg–de Vries equation in H1() concentrate at least the mass of the ground state at the blow-up time. The I-method is used to prove a slightly weaker result in Hs() with 16/17 < s < 1. Under an assumption on the precise blow-up rate, we are able to use similar arguments to prove a more precise analogue of the H1() concentration result over the same range of s.

scholarly journals Ground states for the pseudo-relativistic Hartree equation with external potential

2015 ◽  
Vol 145 (1) ◽  
pp. 73-90 ◽  
Author(s):  
Silvia Cingolani ◽  
Simone Secchi
Keyword(s):  
Ground State ◽  
Schrodinger Equation ◽  
Scalar Potential ◽  
Ground States ◽  
Hartree Equation ◽  
External Potential ◽  
Decay Estimates ◽  
Asymptotic Decay ◽  
Ground State Solutions ◽  
Symmetric Convolution

We prove the existence of positive ground state solutions to the pseudo-relativistic Schrödinger equationwhere N ≥ 3, m > 0, V is a bounded external scalar potential and W is a radially symmetric convolution potential satisfying suitable assumptions. We also provide some asymptotic decay estimates of the found solutions.

scholarly journals Strong Instability of Ground States to a Fourth Order Schrödinger Equation

2017 ◽  
Vol 2019 (17) ◽  
pp. 5299-5315 ◽  
Author(s):  
Denis Bonheure ◽  
Jean-Baptiste Castéras ◽  
Tianxiang Gou ◽  
Louis Jeanjean
Keyword(s):  
Fourth Order ◽  
Ground State ◽  
Blow Up ◽  
Ground States ◽  
Rigorous Results ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Strong Instability ◽  
Blowing Up ◽  
Blowing Up Solutions

Abstract In this note, we prove the instability by blow-up of the ground state solutions for a class of fourth order Schrödinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic nonlinear Schrödinger due to Boulenger and Lenzmann [8] and confirm numerical conjectures from [1–3, 11].

Existence and stability of ground-state solutions of a Schrödinger—KdV system

2003 ◽  
Vol 133 (5) ◽  
pp. 987-1029 ◽  
Author(s):  
John Albert ◽  
Jaime Angulo Pava
Keyword(s):  
Time Dependence ◽  
Ground State ◽  
Nonlinear Waves ◽  
Ground States ◽  
Energy Functional ◽  
Parameter Family ◽  
Existence And Stability ◽  
Image Position ◽  
Range Of Values ◽  
Two Parameter

We consider the coupled Schrödinger–Korteweg–de Vries system which arises in various physical contexts as a model for the interaction of long and short nonlinear waves. Ground states of the system are, by definition, minimizers of the energy functional subject to constraints on conserved functionals associated with symmetries of the system. In particular, ground states have a simple time dependence because they propagate via those symmetries. For a range of values of the parameters α, β, γ, δi, ci, we prove the existence and stability of a two-parameter family of ground states associated with a two-parameter family of symmetries.

Well-posedness and blow-up properties for the generalized Hartree equation

2020 ◽  
Vol 17 (04) ◽  
pp. 727-763
Author(s):  
Anudeep Kumar Arora ◽  
Svetlana Roudenko
Keyword(s):  
Finite Time ◽  
Blow Up ◽  
Hartree Equation ◽  
Energy Threshold ◽  
Local Theory ◽  
Positive Energy ◽  
Small Data ◽  
Well Posedness ◽  
Time Behavior ◽  
Mass Energy

We study the generalized Hartree equation, which is a nonlinear Schrödinger-type equation with a nonlocal potential [Formula: see text]. We establish the local well-posedness at the nonconserved critical regularity [Formula: see text] for [Formula: see text], which also includes the energy-supercritical regime [Formula: see text] (thus, complementing the work in [A. K. Arora and S. Roudenko, Global behavior of solutions to the focusing generalized Hartree equation, Michigan Math J., forthcoming], where we obtained the [Formula: see text] well-posedness in the intercritical regime together with classification of solutions under the mass–energy threshold). We next extend the local theory to global: for small data we obtain global in time existence and for initial data with positive energy and certain size of variance we show the finite time blow-up (blow-up criterion). In the intercritical setting the criterion produces blow-up solutions with the initial values above the mass–energy threshold. We conclude with examples showing currently known thresholds for global vs. finite time behavior.

scholarly journals LOCALIZATION OF THE NUMBER OF PHOTONS OF GROUND STATES IN NONRELATIVISTIC QED

2003 ◽  
Vol 15 (03) ◽  
pp. 271-312 ◽  
Author(s):  
FUMIO HIROSHIMA
Keyword(s):  
Radiation Field ◽  
Ground State ◽  
Fock Space ◽  
Adjoint Operator ◽  
Ground States ◽  
Electron System ◽  
Number Operator ◽  
Position Variable ◽  
Boson Fock Space ◽  
Quantized Radiation Field

One electron system minimally coupled to a quantized radiation field is considered. It is assumed that the quantized radiation field is massless, and no infrared cutoff is imposed. The Hamiltonian, H, of this system is defined as a self-adjoint operator acting on L2 (ℝ3) ⊗ ℱ ≅ L2 (ℝ3; ℱ), where ℱ is the Boson Fock space over L2 (ℝ3 × {1, 2}). It is shown that the ground state, ψg, of H belongs to [Formula: see text], where N denotes the number operator of ℱ. Moreover, it is shown that for almost every electron position variable x ∈ ℝ3 and for arbitrary k ≥ 0, ‖(1 ⊗ Nk/2) ψg (x)‖ℱ ≤ Dk e-δ|x|m+1 with some constants m ≥ 0, Dk > 0, and δ > 0 independent of k. In particular [Formula: see text] for 0 < β < δ/2 is obtained.

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