Strauss's global existence and blowup conjecture for semilinear wave equations

Consider the semilinear wave equation

$$\left\{ \begin{aligned} &\partial_t^2 u-\Delta u=|u|^p, && (t,x)\in\mathbb R^{n+1}_{+},\\ &u(0,\cdot)=u_0(x),\quad \partial_tu(0,\cdot)=u_1(x), \end{aligned} \right.$$

where $p>1$, $n\geq 2$, and $u_i\in C_0^{\infty}(\mathbb R^n)$ for $i=0,1$. Let $p_1(n)$ be the positive root of

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

Strauss's conjecture. If $p>p_1(n)$, then small-data solutions exist globally; if $1<p<p_1(n)$, then small-data solutions blow up in finite time.

The conjecture identifies the critical exponent separating global existence from finite-time blowup for the semilinear wave equation. The source does not state a resolution status or describe the endpoint case $p=p_1(n)$.