  Considering $1+n$ dimensional semilinear wave equations with energy
supercritical powers $p> 1+4/(n-2)$, we obtain global solutions for any initial
data with small norm in $H^{s_c}\times H^{s_c-1}$, under the technical smooth
condition $p>s_c-\bar{s}_0$, with $\bar{s}_0= 1/2+(n-3)/(2\max(n-1-p,n-3))$ and
$s_c=n/2-2/(p-1)$. In particular, combined with previous works, our results
give a complete verification of the Strauss conjecture, up to space dimension
$9$. The higher dimensional case, $n\ge 10$, seems to be unreachable, in view
of the wellposed theory in $H^s$.
