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On local existence for nonlinear wave equations satisfying variable coefficient null conditions. (English) Zbl 0792.35125

Quasilinear wave equations are considered on \(M^ n\), \(M^ n\) being \(\mathbb{R}^ n\) or a \(C^ \infty\) compact boundaryless manifold of dimension \(n \geq 2\), with principal linear part having coefficients depending on time \(t\) and space variable \(x\). Space-time estimates for the associated null forms are presented, and a local existence theorem in lower order Sobolev spaces is proved. This extends the results of S. Klainerman and M. Machedon [Commun. Pure Appl. Math. 46, 1221-1268 (1993)] where \(M^ n = \mathbb{R}^ n\), and the operator equals the d’Alembert operator. Fourier integral operator techniques are the main tool.
Reviewer: R.Racke (Konstanz)

MSC:

35L70 Second-order nonlinear hyperbolic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs
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References:

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