Global solutions and finite time blow up for damped Klein-Gordon equation

2013 ◽  
Vol 33 (3) ◽  
pp. 643-652 ◽  
Author(s):  
Runzhang XU ◽  
Yunhua DING
Keyword(s):  
Finite Time ◽  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
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.

scholarly journals Revised concavity method and application to Klein-Gordon equation

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 831-839 ◽  
Author(s):  
M. Dimova ◽  
N. Kolkovska ◽  
N. Kutev
Keyword(s):  
Differential Inequality ◽  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Initial Energy ◽  
Necessary And Sufficient Condition ◽  
Positive Initial Energy ◽  
Klein Gordon ◽  
Necessary And Sufficient ◽  
Klein Gordon Equation

A revised version of the concavity method of Levine, based on a new ordinary differential inequality, is proposed. Necessary and sufficient condition for nonexistence of global solutions of the inequality is proved. As an application, finite time blow up of the solution to Klein-Gordon equation with arbitrary positive initial energy is obtained under very general structural conditions.

scholarly journals Sharp Condition for Global Existence and Blow-Up on Klein-Gordon Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
Zhao Junsheng ◽  
Li Shufeng
Keyword(s):  
Gordon Equation ◽  
Blow Up ◽  
Global Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Potential Wells ◽  
Sharp Condition ◽  
Klein Gordon ◽  
Initial Boundary Value ◽  
Klein Gordon Equation

We study the initial boundary value problem of the nonlinear Klein-Gordon equation. First we introduce a family of potential wells. By using them, we obtain a new existence theorem of global solutions and show the blow-up in finite time of solutions. Especially the relation between the above two phenomena is derived as a sharp condition.

Global solutions of quasilinear wave-Klein–Gordon system in two-space dimension: Technical tools

2017 ◽  
Vol 14 (04) ◽  
pp. 591-625 ◽  
Author(s):  
Yue Ma
Keyword(s):  
Wave Equation ◽  
Vector Field ◽  
Space Dimension ◽  
Gordon Equation ◽  
Global Solutions ◽  
Field Method ◽  
Time Dimension ◽  
Klein Gordon ◽  
Vector Field Method ◽  
Klein Gordon Equation

In this paper and its successor, we make an application of the hyperboloidal foliation method in [Formula: see text] space-time dimension. After the establishment of some technical tools in this paper, we will prove further the global existence of small regular solution to a class of hyperbolic system composed by a wave equation and a Klein–Gordon equation with null couplings. Our method belongs to vector field method and, more precisely, is a combination of the normal form and the hyperboloidal foliation method.

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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