scholarly journals Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions

2001 ◽  
Vol 53 (4) ◽  
pp. 875-912 ◽  
Author(s):  
Hideo KUBO ◽  
Kôji KUBOTA
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equations ◽  
Classical Solutions ◽  
Semilinear Wave Equations

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 LOWER BOUND OF THE LIFESPAN OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS IN AN EXTERIOR DOMAIN

2013 ◽  
Vol 10 (02) ◽  
pp. 199-234
Author(s):  
SOICHIRO KATAYAMA ◽  
HIDEO KUBO
Keyword(s):  
Cauchy Problem ◽  
Lower Bound ◽  
Exterior Domain ◽  
Wave Equations ◽  
Symmetric Case ◽  
Classical Solutions ◽  
Semilinear Wave Equations ◽  
The Cauchy Problem ◽  
Three Space ◽  
Dimensional Domain

We consider the Cauchy–Dirichlet problem for semilinear wave equations in a three space-dimensional domain exterior to a bounded and non-trapping obstacle. We obtain a detailed estimate for the lower bound of the lifespan of classical solutions when the size of initial data tends to zero, in a similar spirit to that of the works of John and Hörmander where the Cauchy problem was treated. We show that our estimate is sharp at least for radially symmetric case.

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