Global dynamics of a supercritical wave equation in a large data regime

We prove the existence of global solutions to the nonlinear wave equation in ${\mathbb{R}}^{1+3}$ $${\mathrm{\Phi }}_{tt}-\mathrm{\Delta }\mathrm{\Phi }\pm \mathrm{\Phi }\mathrm{\mid }\mathrm{\Phi }{\mathrm{\mid }}^{p-1}=0$$ in the energy-supercritical regime $p>5$, for a class of large initial data. Our initial data can be decomposed into two pieces, one which is dispersed in the sense of having large $L^2$ norm, while the other piece takes a localised short-pulse form. Consequently, we can obtain global existence for a class of initial data which is large in every homogeneous Sobolev norm ${\dot{H}}_x^s$ with $s\ge 0$.