  This paper is concerned with the Cauchy problem for the semilinear wave
equation: $u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty)$, where the
space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here,
the Cauchy data are non-zero and non-compactly supported. Our results on the
blow-up of positive radial solutions (not necessarily radial in low dimensions
$n=2, 3$) generalize and extend the results of Takamura(1995) and Takamura,
Uesaka and Wakasa(2011). The main technical difficulty in the paper lies in
obtaining the lower bounds for the free solution when both initial position and
initial velocity are non-identically zero in even space dimensions.
