We are concerned with classical solutions to the initial value problem for $\square u=|u|^p$ or $\square u=|u_t|^p$ in $\Bbb R^n\times[0,\infty)$ with \lq\lq small" data. If the data have compact support, it is partially known that there is a critical number $p_0(n)$ such that most solutions blow-up in finite time for $1<p\le p_0(n)$ and a global solution exists for $p>p_0(n)$. In this paper, we shall show for all $n\ge2$ that, if the support of data is noncompact, there are blowing-up solutions even for $p>p_0(n)$ because of the \lq\lq bad" spatial decay of the initial data. Moreover, critical decays for each equation are conjectured. The proof lies in the pointwise estimates of the fundamental solution of $\square$.
Hiroyuki Takamura. "Blow-up for semilinear wave equations with slowly decaying data in high dimensions." Differential Integral Equations 8 (3) 647 - 661, 1995. https://doi.org/10.57262/die/1369316512