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-trapping obstacles as well as any obstacle that has exponential local decay rate of energy for H data for the linear equation (see (1.4) below). This hypothesis is fulfilled in the non-trapping case where there is actually exponential local decay of energy [19] with no loss of derivatives. This hypothesis (1.4) is also known to hold in several examples involving hyperbolic trapped rays. For instance, our results apply to situations where the obstacle is a finite union of convex bodies with smooth boundary (see [7], [8]). In addition to improving the hypotheses on the obstacles, we can also improve considerably on the decay assumptions on the initial data at infinity compared to the results in [9] which were obtained by the conformal method. Lastly, we are able handle non-diagonal systems involving multiple wave speeds.