Full blow-up range for co-rotaional wave maps to surfaces of revolution

Abstract

We construct blow-up solutions of the energy critical wave map equation on ${\mathbb{R}}^{2+1}\to \mathcal{N}$ with polynomial blow-up rate ($t^{-1-\hat{I}½}$ for blow-up at $t=0$) in the case when $\mathcal{N}$ is a surface of revolution. Here we extend the blow-up range found by Cârstea ($\hat{I}½>\frac{1}{2}$) based on the work by Krieger, Schlag and Tataru to $\hat{I}½>0$. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere.