Chen–Reissig's threshold conjecture for semilinear wave equations

Let $p_S$ denote the Strauss exponent in dimension three, and consider the Cauchy problem for the semilinear wave equation with nonlinearity

$$F(u)=|u|^{p_S}\mu(|u|),$$

where $\mu:[0,\infty)\to[0,\infty)$ is continuous and increasing with $\mu(0)=0$. Define

$$C_{\mathrm{Str}}=\lim_{\lambda\to0^+}\mu(\lambda)\left(\log\frac1\lambda\right)^{\frac1{p_S}}.$$

Chen–Reissig's conjecture. If $C_{\mathrm{Str}}>0$, then the solution blows up at finite time for some initial data; if $C_{\mathrm{Str}}=0$, then small-data global existence holds. This conjecture seeks the sharp threshold between blowup and global existence for nonlinearities that modify the Strauss power by a modulus of continuity. The cited logarithmic results establish only an almost-sharp threshold, so the general criterion remains unresolved.