A class of large global solutions for the Wave--Map equation

In this paper we consider the equation for equivariant wave maps from $R^{3+1}$ to $S^3$ and we prove global in forward time existence of certain $C^{\mathrm{\infty }}$-smooth solutions which have infinite critical Sobolev norm ${\dot{H}}^{\frac{3}{2}}(R^3)\times {\dot{H}}^{\frac{1}{2}}(R^3)$. Our construction provides solutions which can moreover satisfy the additional size condition $\mathrm{\parallel }u(0,\cdot ){\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\mid }x\mathrm{\mid }\ge 1)}>M$ for arbitrarily chosen $M>0$. These solutions are also stable under suitable perturbations. Our method is based on a perturbative approach around suitably constructed approximate self–similar solutions.