Global solutions of quasilinear wave equations
نویسنده
چکیده
has a global solution for all t ≥ 0 if initial data are sufficiently small. Here the curved wave operator is ̃g = g ∂α∂β, where we used the convention that repeated upper and lower indices are summed over α, β = 0, 1, 2, 3, and ∂0 = ∂/∂t, ∂i = ∂/∂x i, i = 1, 2, 3. We assume that gαβ(φ) are smooth functions of φ such that gαβ(0)= mαβ , where m00=−1, m11= m22= m33=1 and mαβ= 0, if α 6=β. The result holds for vector valued φ, in particular for the principal part of Einstein’s equations; φαβ= gαβ−mαβ. This result was conjectured in [L2] where it was also shown in the spherically symmetric case for
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