LAUSR

Abstract We study the long time existence for the small-amplitude semilinear wave equations with mixed nonlinearities of the form c 1 | u t | p + c 2 u 2 with p ≥ 2 , when the spatial dimension is four. By exploiting the local energy estimates and their recent variants, we prove the almost global existence up to exp ⁡ ( c e − 2 ) , which is sharp in general. For the case p ∈ ( 2 , 3 ) , due to technical reason, we need to assume the initial data to be radial.