*Accepted Paper*

**Inserted:** 20 apr 2015

**Last Updated:** 12 aug 2015

**Journal:** Transactions of the AMS

**Year:** 2015

**Abstract:**

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.

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