Generalized Strauss conjecture for semilinear wave equations on R³

In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation ∂ₜ² u-ΔR³u=|u|pSμ(|u|)\ ,u(0,x)=ε u₀,\ uₜ(0,x)=ε u₁\ , where pS=1+√2 is the Strauss critical index in $n=3$, and μ is a modulus of continuity. Inspired by Chen, Reissig{Chen_2024} and Ebert, Girardi, Reissig{MR4163528}, we investigate the sharp condition of μ as the threshold between the global existence and blow up with small data. We obtain the almost sharp results in this paper, which in particular disproves the conjecture in {Chen_2024}.