Global Solutions with Small Initial Data to Semilinear Wave Equations with Energy Supercritical Powers

Considering $1+n$ dimensional semilinear wave equations with energy supercritical powers $p>1+4\mathrm{/}(n-2)$, 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 ${\bar{s}}_0=1\mathrm{/}2+(n-3)\mathrm{/}(2\max (n-1-p,n-3))$ and $s_c=n\mathrm{/}2-2\mathrm{/}(p-1)$. 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 $H^s$.