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

In this paper we consider the equation for equivariant wave maps from $\R^{3+1}$ to $\Sf^3$ 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, strongly inspired by \cite{krsc}, is based on a perturbative approach around suitably constructed approximate self--similar solutions.

Published in:

Transactions- American Mathematical Society

Publisher:

American Mathematical Society

ISSN:

0002-9947

Laboratories:

Record created 2015-04-21, last modified 2018-01-28