Local energy decay for linear wave equations with non-compactly supported initial data

2004 ◽  
Vol 27 (16) ◽  
pp. 1881-1892 ◽  
Author(s):  
Ryo Ikehata
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Energy Decay ◽  
Linear Wave ◽  
Local Energy ◽  
Compactly Supported ◽  
Linear Wave Equations

LOCAL EXISTENCE FOR SEMILINEAR WAVE EQUATIONS AND APPLICATIONS TO YANG–MILLS EQUATIONS

2005 ◽  
Vol 02 (01) ◽  
pp. 61-76
Author(s):  
YUNG-FU FANG
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Nonlinear Term ◽  
A Priori ◽  
Local Existence ◽  
A Priori Estimate ◽  
Linear Wave ◽  
Yang Mills ◽  
Linear Wave Equations ◽  
Local Result

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H1+∊ and H∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

scholarly journals SOLUTIONS OF QUASI-LINEAR WAVE EQUATIONS POLYHOMOGENEOUS AT NULL INFINITY IN HIGH DIMENSIONS

2011 ◽  
Vol 08 (02) ◽  
pp. 269-346 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
ROGER TAGNE WAFO
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Small Data ◽  
Linear Wave ◽  
High Dimensions ◽  
Null Infinity ◽  
Linear Wave Equations

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein–Maxwell equations in space-time dimensions n + 1 ≥ 7. Similarly we prove propagation of polyhomogeneity in dimensions n + 1 ≥ 9. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein–Maxwell equations evolving out of initial data which are stationary outside of a ball.

scholarly journals Scattering for critical wave equations with variable coefficients

2021 ◽  
pp. 1-19
Author(s):  
Shi-Zhuo Looi ◽  
Mihai Tohaneanu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Variable Coefficients ◽  
Energy Decay ◽  
Time Dependent ◽  
Linear Wave ◽  
Decay Estimates ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Semilinear Wave Equation

Abstract We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$ , but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$ .

scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

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