Differential and Integral Equations

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

Sigmund Selberg

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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