Concerning the Strauss conjecture on asymptotically Euclidean manifolds

In this paper we verify the Strauss conjecture for semilinear wave equations on asymptotically Euclidean manifolds when n = 3 , 4 . We also give an almost sharp lifespan for the subcritical case 2 ⩽ p < p c when n = 3 . The main ingredients include a Keel–Smith–Sogge type estimate with 0 < μ < 1 / 2 and weighted Strichartz estimates of order two.