A class of large global solutions for the wave-map equation

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 H^{3/2}(R^3)x H^{3/2}(R^3). Our construction provides solutions which can moreover satisfy the additional size condition |u(0,·)| L^infty(|x|geq1) > M for arbitrarily chosen M > 0. These solutions are also stable under suitable perturbations. Our method, strongly inspired by work of Krieger and Schlag, is based on a perturbative approach around suitably constructed approximate self-similar solutions