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 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 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.

Time decay for the nonlinear Klein-Gordon equation

1968 ◽  
Vol 306 (1486) ◽  
pp. 291-296 ◽  
Keyword(s):  
Gordon Equation ◽  
Initial Energy ◽  
Local Energy ◽  
Time Decay ◽  
Systems Of Equations ◽  
Second Derivatives ◽  
Klein Gordon ◽  
Piecewise Continuous ◽  
Klein Gordon Equation

It is shown that solutions of the nonlinear Klein-Gordon equation u tt - ∆ u + mu + P '( u ) = 0 decay to zero in the local L 2 mean if the initial energy is bounded provided s P ') s ) - 2 P ( s ) ≥ a P ( s ) ≥ 0 with a > 0. The local energy also decays. The proof is based on manipulating energy identities and re­quires that u have continuouś first derivatives and piecewise continuous second derivatives. The proof is also applicable to certain systems of equations.

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.

scholarly journals General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

2014 ◽  
Vol 2014 ◽  
pp. 1-21 ◽  
Author(s):  
Wenjun Liu ◽  
Gang Li ◽  
Linghui Hong
Keyword(s):  
Blow Up ◽  
Global Solutions ◽  
Initial Energy ◽  
General Decay ◽  
Strong Damping ◽  
Kirchhoff Type ◽  
Positive Initial Energy ◽  
Relaxation Functions ◽  
Viscoelastic Equations ◽  
Arbitrarily Positive Initial Energy

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.

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