Scattering for semilinear wave equations with small data in three space dimensions

1988 ◽  
Vol 198 (2) ◽  
pp. 277-289 ◽  
Author(s):  
Hartmut Pecher
Keyword(s):  
Wave Equations ◽  
Small Data ◽  
Semilinear Wave Equations ◽  
Three Space

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 Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

scholarly journals Critical exponent of Fujita-type for semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime

2021 ◽  
Author(s):  
Marcelo Ebert ◽  
Jorge Marques
Keyword(s):  
Wave Equation ◽  
Critical Exponent ◽  
Wave Equations ◽  
Small Data ◽  
Semilinear Wave Equations

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

Sign in / Sign up

Export Citation Format

Share Document