In this paper we consider the equation for equivariant wave maps from
R3+1 to S3 and we prove global in forward time existence of certain
C∞-smooth solutions which have infinite critical Sobolev norm
H˙23(R3)×H˙21(R3). Our construction
provides solutions which can moreover satisfy the additional size condition
∥u(0,⋅)∥L∞(∣x∣≥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