Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}

For the radial energy-supercritical nonlinear wave equation $$\mathrm{\square }u=-u_{tt}+\mathrm{△}u=\pm u^7$$ on ${\mathbb{R}}^{3+1}$, we prove the existence of a class of global in forward time $C^{\mathrm{\infty }}$-smooth solutions with infinite critical Sobolev norm ${\dot{H}}^{\frac{7}{6}}({\mathbb{R}}^3)\times {\dot{H}}^{\text{f}16}({\mathbb{R}}^3)$. Moreover, these solutions are stable under suitably small perturbations . We also show that for the {\em defocussing} energy supercritical wave equation, we can construct such solutions which moreover satisfy the size condition $|u(0, \cdot)|_{L_x^\infty(|x|\geq 1)}>M$forarbitrarilyprescribed$M>0$. These solutions are stable under suitably small perturbations. Our method proceeds by regularization of self-similar solutions which are smooth away from the light-cone but singular on the light-cone. The argument crucially depends on the supercritical nature of the equation. Our approach should be seen as part of the program initiated in \cite{KrSchTat1}, \cite{KrSchTat2}, \cite{DoKri}.