Generalized Strichartz Estimates on Perturbed Wave Equation and Applications on Strauss Conjecture

The purpose of this paper is to show a general Strichartz estimate for certain perturbed wave equation under known local energy decay estimates, and as application, to get the Strauss conjecture for several convex obstacles in n = 3, 4. Our results improve on earlier work in Hidano, Metcalfe, Smith, Sogge and Zhou [14]. First, and most important, we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates(see (1.2) below). This hypothesis is fulfilled in many cases in the non-trapping case when there is local decay of energy with no loss of derivatives( see [32],[21],[30],[3],[23]), (1.2) is also known to hold in several examples involving hyperbolic trapped rays(see [15],[16],[9]). In addition to improving the hypotheses on the obstacles, we give the obstacle version of sharp life span for semilinear wave equations when n = 3, p < pc, by using a real interpolation between KSS estimate and endpoint Trace Lemma, and by getting a corresponding finite time Strichartz estimates(see section 3). Lastly, we are able to use the general Strichartz estimates we have gained to get the Strauss conjecture for some perturbed semilinear wave equations with trapped rays when n = 3, 4(see Section 4). We consider wave equations on an exterior domain Ω ⊂ R: