Global existence for quasilinear wave equations close to Schwarzschild

Hyperbolic Trapped Rays and Global Existence of Quasilinear Wave Equations

The purpose of this paper is to give a simple proof of global existence for quadratic quasilinear Dirichlet-wave equations outside of a wide class of compact obstacles in the critical case where the spatial dimension is three. Our results improve on earlier ones in Keel, Smith and Sogge [9] in several ways. First, and most important, we can drop the star-shaped hypothesis and handle non-trappin...

Global Existence for Quasilinear Dispersive Equations

Global Existence for Quasilinear Dispersive Equations Ii

is invertible. Here we treat the case of inputs coming from free waves located far away from the origin. In order to simplify the presentation, we will only consider inputs that would have come from our first guess (that is |x|f is bounded in L). Hence, we now consider the case when at least one function f or g is supported away from the origin and assume that for all y1 in the support of f and...

Global Existence for High Dimensional Quasilinear Wave Equations Exterior to Star-shaped Obstacles

1. Introduction. The purpose of this article is to study long time existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. In particular, we seek to prove exterior domain analogs of the four dimensional results of [5] where the nonlinearity is permitted to depend on the solution not just its first and second derivatives. Previous proofs in exterior domains o...

Global solutions of quasilinear wave equations

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 resul...