Global Existence of Small Amplitude Solutions to Nonlinear Klein-Gordon Equations in Four Space-Time Dimensions

1987 ◽  
pp. 75-89 ◽  
Author(s):  
S. Klainerman
Keyword(s):  
Global Existence ◽  
Small Amplitude ◽  
Space Time ◽  
Klein Gordon

Global solutions of quasilinear wave-Klein–Gordon system in two-space dimension: Completion of the proof

2017 ◽  
Vol 14 (04) ◽  
pp. 627-670 ◽  
Author(s):  
Yue Ma
Keyword(s):  
Global Existence ◽  
Space Dimension ◽  
Global Solutions ◽  
Space Time ◽  
Time Dimension ◽  
Regular Solutions ◽  
Complete Proof ◽  
Klein Gordon

Based on the first part, we give a complete proof of the global existence of small regular solutions to a type of quasilinear wave-Klein–Gordon system with null couplings in [Formula: see text] space-time dimension.

scholarly journals Global existence of small amplitude solutions to one-dimensional nonlinear Klein–Gordon systems with different masses

2015 ◽  
Vol 12 (04) ◽  
pp. 745-762 ◽  
Author(s):  
Donghyun Kim
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Small Amplitude ◽  
Space Dimension ◽  
Structural Condition ◽  
Cauchy Data ◽  
One Dimensional ◽  
Compactly Supported ◽  
The Cauchy Problem ◽  
Klein Gordon

We study the Cauchy problem for systems of cubic nonlinear Klein–Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the rate [Formula: see text] in [Formula: see text], [Formula: see text] as [Formula: see text] tends to infinity even in the case of mass resonance, if the Cauchy data are sufficiently small, smooth and compactly supported.

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

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