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 Blowup for the Cubic Nonlinear Klein-Gordon Equations in Three Space Dimensions

2010 ◽  
Vol 10 (2) ◽  
Author(s):  
Jian Zhang ◽  
Zaihui Gan ◽  
Boling Guo
Keyword(s):  
Global Existence ◽  
Invariant Manifolds ◽  
Gordon Equation ◽  
Global Solutions ◽  
The Other ◽  
Cubic Nonlinearity ◽  
Sharp Threshold ◽  
Klein Gordon ◽  
Three Space ◽  
Klein Gordon Equation

AbstractIn this paper, we apply a cross-constrained variational method to study the classic nonlinear Klein-Gordon equation with cubic nonlinearity in three space dimensions. By constructing a type of cross-constrained variational problem and establishing the so-called cross invariant manifolds, we obtain a sharp threshold for blowup and global existence of the solution to the equation under study which is different from that in [10] . On the other hand, we give an answer to the question that how small the initial data have to be for the global solutions to exist.

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 Global Existence and Blow-Up of Solutions for Nonlinear Klein-Gordon Equation with Damping Term and Nonnegative Potentials

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Wen-Yi Huang ◽  
Wen-Li Chen
Keyword(s):  
Global Existence ◽  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Invariant Sets ◽  
Potential Well ◽  
Damping Term ◽  
Potential Wells ◽  
Klein Gordon ◽  
Klein Gordon Equation

This paper is concerned with the nonlinear Klein-Gordon equation with damping term and nonnegative potentials. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we obtain a new existence theorem of global solutions and a blow-up result for solutions in finite time.

scholarly journals Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chenglin Wang ◽  
Jian Zhang
Keyword(s):  
Global Existence ◽  
Schrödinger Equation ◽  
Variational Method ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Manifolds ◽  
Nonlinear Schrodinger Equation ◽  
Sharp Condition ◽  
Nonlinear Schrödinger ◽  
Partial Confinement

<p style='text-indent:20px;'>In this paper, we study the nonlinear Schrödinger equation with a partial confinement. By applying the cross-constrained variational arguments and invariant manifolds of the evolution flow, the sharp condition for global existence and blowup of the solution is derived.</p>

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.

Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

2018 ◽  
Vol 15 (04) ◽  
pp. 755-788
Author(s):  
Vladimir Georgiev ◽  
Sandra Lucente
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Ground State ◽  
Nonlinear Term ◽  
Blow Up ◽  
Critical Energy ◽  
Ground State Solution ◽  
Energy Space ◽  
Nirenberg Inequality ◽  
Klein Gordon

We study the dynamics for the focusing nonlinear Klein–Gordon equation, [Formula: see text] with positive radial potential [Formula: see text] and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on [Formula: see text]. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

Generalized Invariant Sets for the Boltzmann Equation

1997 ◽  
Vol 07 (04) ◽  
pp. 457-476 ◽  
Author(s):  
T. Goudon
Keyword(s):  
Boltzmann Equation ◽  
Initial Data ◽  
Global Existence ◽  
Existence Of Solutions ◽  
Invariant Set ◽  
Invariant Sets ◽  
Soft Or ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
The Boltzmann Equation

We are interested in the initial value problem for the Boltzmann equation, when the initial data u0 belongs to a set B0 = {δ0m1 (0,x,v) ≤ u0(x,v) ≤ C0m2 (0,x,v)} where m1, m2 are traveling Maxwellians. We consider soft or Maxwell's interactions with cutoff (7/3 < s ≤ 5) and C0 smaller than a bound depending on the coefficients of m2. We obtain global existence of solutions remaining in a "generalized invariant set" Bt ⊂ B∞, characterized by these particular states.

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