Generalized Strauss conjecture for semilinear wave equations on ${\mathbb{R}}^3$

In this manuscript, we focus on the more delicate nonlinearity of the semilinear wave equation $${\mathrm{\partial }}_t^{2}u-{\mathrm{\Delta }}_{{\mathbb{R}}^3}u=\mathrm{\mid }u{\mathrm{\mid }}^{p_S}\mu (\mathrm{\mid }u\mathrm{\mid }),u(0,x)=\epsilon u_0,u_t(0,x)=\epsilon u_1,$$ where $p_S=1+\sqrt{2}$ is the Strauss critical index in $n=3$, and $\mu$ is a modulus of continuity. Inspired by Chen, Reissig\cite{Chen_2024} and Ebert, Girardi, Reissig\cite{MR4163528}, we investigate the sharp condition of $\mu$ 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 \cite{Chen_2024}.