scholarly journals ASYMPTOTIC POINTWISE BEHAVIOR FOR SYSTEMS OF SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

2012 ◽  
Vol 09 (02) ◽  
pp. 263-323 ◽  
Author(s):  
SOICHIRO KATAYAMA
Keyword(s):  
Asymptotic Behavior ◽  
Global Existence ◽  
Small Amplitude ◽  
Wave Equations ◽  
Global Solutions ◽  
Small Data ◽  
Sufficient Condition ◽  
Null Condition ◽  
Semilinear Wave Equations ◽  
Three Space

In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.

scholarly journals Global Existence for a System of Quasi-Linear Wave Equations in 3D Satisfying the Weak Null Condition

2018 ◽  
Vol 2020 (1) ◽  
pp. 39-70
Author(s):  
Kunio Hidano ◽  
Kazuyoshi Yokoyama
Keyword(s):  
Global Existence ◽  
Wave Equations ◽  
Linear Wave ◽  
Null Condition ◽  
Standard Argument ◽  
The Cauchy Problem ◽  
Weight Method ◽  
Linear Wave Equations ◽  
Three Space ◽  
Original Theorem

Abstract We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the presence of some quadratic nonlinear terms which do not satisfy the null condition. Due to the presence of such quadratic terms, the standard argument no longer works for the proof of global existence. To get over this difficulty, we extend the ghost weight method of Alinhac so that it works for the system under consideration. The original theorem of Alinhac for the scalar unknowns is also refined.

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