The critical-exponent conjecture for semilinear wave equations with noneffective damping

Consider the semilinear wave equation

$$\partial_t^2u-\Delta u+\frac{\mu}{1+t}\partial_tu=|u|^p,$$

with $0<\mu<2$, where $p_S(n+\mu)$ denotes the positive root of

$$(n-1+\mu)p^2-(n+1+\mu)p-2=0.$$

Critical-exponent conjecture. The critical exponent of the equation is

$$p_c=p_S(n+\mu).$$

The preceding theorem proves finite-time blow-up for $1<p<p_S(n+\mu)$, while related results establish criticality in some cases such as $\mu=2$. For $0<\mu<2$, whether $p_S(n+\mu)$ is the critical exponent remains open.