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In this paper we show a general Strichartz estimate for certain perturbedwave equation, and here we can drop the nontrapping hypothesis and handletrapping obstacles with some loss of derivatives for data in the local energydecay estimates. Then we give the obstacle version of sharp life span forsemilinear wave equations when $n=3,p<p_c$, by using a real interpolationmethod, and by getting a corresponding finite time Strichartz estimates(seesection 3). Finally, as an application of the general Strichartz estimates wehave gained, we get the Strauss conjecture for semilinear wave equations withseveral convex obstacles when $n=3,4$(see Section 4).

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