Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture

Abstract

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).