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On an estimate for the wave equation and applications to nonlinear problems. (English) Zbl 1032.35121

This paper considers the solution of the Cauchy problem \[ \begin{cases}\partial^2_t u-\Delta u=F &\text{in }{\mathbb R}^{n+1}\\ u_{|t=0}=f,\;\partial_t u_{|t=0}=g &\text{ in }{\mathbb R}^n.\end{cases} \] Some estimates are proved for \(u\) in certain classes of Banach spaces whose norms depend only on the size of the space-time Fourier transform. These estimates are local in time and are similar to those obtained by J. Bourgain for Schrödinger and KdV equations [Geom. Funct. Anal. 3, 107-156 (1993; Zbl 0787.35097); Geom. Funct. Anal. 3, 209-262 (1993; Zbl 0787.35098)]. Applications of the estimates obtained here are performed in proving the well-posedness of some Cauchy problems for nonlinear wave equations; other applications can be found in [S. Klainerman and the author, Commun. Contemp. Math. 4, 223-295 (2002; Zbl 1146.35389)].

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
35L05 Wave equation
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