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/(n-2) $, we obtained 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 well-posed theory in $ H^s $.