Sharp Threshold of the Gross-Pitaevskii Equation with Trapped Dipolar Quantum Gases

2013 ◽  
Vol 56 (2) ◽  
pp. 378-387 ◽  
Author(s):  
Li Ma ◽  
Jing Wang
Keyword(s):  
Global Existence ◽  
Variational Problem ◽  
Invariant Manifold ◽  
Finite Time ◽  
Blow Up ◽  
Quantum Gases ◽  
Pitaevskii Equation ◽  
Sharp Threshold ◽  
Unstable Regime

AbstractIn this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.

A cross-constrained variational problem for the generalized Davey–Stewartson system

2009 ◽  
Vol 52 (1) ◽  
pp. 67-77
Author(s):  
Zaihui Gan ◽  
Jian Zhang
Keyword(s):  
Global Existence ◽  
Variational Problem ◽  
Standing Wave ◽  
Ground State ◽  
Invariant Manifolds ◽  
Blow Up ◽  
Standing Waves ◽  
Sharp Threshold ◽  
Image Position

AbstractWe study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson systemin ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

scholarly journals Sharp thresholds of blow-up and global existence for the Schrödinger equation with combined power-type and Choquard-type nonlinearities

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yongbin Wang ◽  
Binhua Feng
Keyword(s):  
Global Existence ◽  
Finite Time ◽  
Critical Case ◽  
Blow Up ◽  
Critical Mass ◽  
Choquard Equation ◽  
Power Type ◽  
Scale Invariant ◽  
Nonlinear Schrödinger ◽  
Sharp Thresholds

AbstractIn this paper, we consider the sharp thresholds of blow-up and global existence for the nonlinear Schrödinger–Choquard equation $$ i\psi _{t}+\Delta \psi =\lambda _{1} \vert \psi \vert ^{p_{1}}\psi +\lambda _{2}\bigl(I _{\alpha } \ast \vert \psi \vert ^{p_{2}}\bigr) \vert \psi \vert ^{p_{2}-2}\psi . $$iψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ. We derive some finite time blow-up results. Due to the failure of this equation to be scale invariant, we obtain some sharp thresholds of blow-up and global existence by constructing some new estimates. In particular, we prove the global existence for this equation with critical mass in the $L^{2}$L2-critical case. Our obtained results extend and improve some recent results.

scholarly journals CROSS-CONSTRAINED VARIATIONAL PROBLEM AND THE NON-LINEAR KLEIN–GORDON EQUATIONS

2008 ◽  
Vol 50 (3) ◽  
pp. 467-481 ◽  
Author(s):  
ZAIHUI GAN ◽  
JIAN ZHANG
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Variational Method ◽  
Variational Problem ◽  
Invariant Manifolds ◽  
Invariant Sets ◽  
Sharp Threshold ◽  
Non Linear ◽  
Klein Gordon ◽  
Three Space

AbstractIn this paper, we put forward a cross-constrained variational method to study the non-linear Klein–Gordon equations with an inverse square potential in three space dimensions. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we establish some new types of invariant sets for the equation and derive a sharp threshold of blowup and global existence for its solution. Finally, we give an answer to the question how small the initial data are for the global solution to exist.

Global Existence and Blow-Up in Finite Time of Solutions to One Kind of Reaction-Diffusion Educations with Neumann Boundary Conditions

2014 ◽  
Vol 971-973 ◽  
pp. 1017-1020
Author(s):  
Jun Zhou Shao ◽  
Ji Jun Xu
Keyword(s):  
Global Existence ◽  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Blowing Up ◽  
Processing Techniques

This paper deals with the properties of one kind of reaction-diffusion equations with Neumann boundary conditions based on the comparison principles. The relations of parameter and the situation of the coupled about equations are used to construct the global existent super-solutions and the blowing-up sub-solutions, and then we obtain the conditions of the global existence and blow-up in finite time solutions with the processing techniques of inequality.

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