On the Cauchy and Cauchy–Darboux problems for semilinear wave equations

2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Otar Jokhadze ◽  
Sergo Kharibegashvili
Keyword(s):  
Wave Equations ◽  
Global Solutions ◽  
Local Solvability ◽  
Continuous Functions ◽  
Semilinear Wave Equations

AbstractThe Cauchy and Cauchy–Darboux problems for semilinear wave equations in the class of continuous functions are investigated. The questions of existence, uniqueness and nonexistence of global solutions of the problems are considered. The local solvability of the problems is also discussed.

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.

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