Differential and Integral Equations

On an estimate for the wave equation and applications to nonlinear problems

Sigmund Selberg

Abstract

We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\mathbb R^{1+n}$ in a class of Banach spaces whose norms depend only on the size of the space--time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman--Machedon [4, 5]. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations, which was used extensively in the recent article [8].

Article information

Source
Differential Integral Equations, Volume 15, Number 2 (2002), 213-236.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060873

Mathematical Reviews number (MathSciNet)
MR1870470

Zentralblatt MATH identifier
1032.35121

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B35: Stability

Citation

Selberg, Sigmund. On an estimate for the wave equation and applications to nonlinear problems. Differential Integral Equations 15 (2002), no. 2, 213--236. https://projecteuclid.org/euclid.die/1356060873