  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
$C^\infty$-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
$\|u(0, \cdot)\|_{L^\infty(|x|\geq 1)}>M$ 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.
