BILINEAR ESTIMATES AND APPLICATIONS TO GLOBAL WELL-POSEDNESS FOR THE DIRAC–KLEIN–GORDON EQUATION ON ℝ1+1

2013 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
TIMOTHY CANDY
Keyword(s):  
Global Existence ◽  
Gordon Equation ◽  
Well Posedness ◽  
Bilinear Estimates ◽  
Dirac Equations ◽  
Klein Gordon ◽  
Nonlinear Dirac Equations ◽  
Klein Gordon Equation

We prove new bilinear estimates for the [Formula: see text] spaces which are optimal up to endpoints. These estimates are often used in the theory of nonlinear Dirac equations on ℝ1+1. As an application, by using the I-method of Colliander, Keel, Staffilani, Takaoka and Tao, we extend the work of Tesfahun on global existence below the charge class for the Dirac–Klein–Gordon equation on ℝ1+1.

scholarly journals Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

2011 ◽  
Vol 30 (3) ◽  
pp. 573-621 ◽  
Author(s):  
M. Keel ◽  
◽  
Tristan Roy ◽  
Terence Tao ◽  
◽  
...  
Keyword(s):  
Gordon Equation ◽  
Energy Norm ◽  
Well Posedness ◽  
Klein Gordon ◽  
Klein Gordon Equation

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.

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.

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