Almost global existence for semilinear wave equations with mixed nonlinearities in four space dimensions

We study the long time existence for the small-amplitude semilinear wave equations with mixed nonlinearities of the form c1|ut|p+c2u2 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⁡(cε−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.