A Construction of Blow up Solutions for Co-rotational Wave Maps

Abstract

The existence of co-rotational finite time blow up solutions to the wave map problem from R^{2+1} into N, where N is a surface of revolution with metric d\rho^2+g(\rho)^2 d\theta^2, g an entire function, is proven. These are of the form u(t,r)=Q(\lambda(t)t)+R(t,r), where Q is a time independent solution of the co-rotational wave map equation -u_{tt}+u_{rr}+r^{-1}u_r=r^{-2}g(u)g'(u), \lambda(t)=t^{-1-\nu}, \nu>1/2 is arbitrary, and R is a term whose local energy goes to zero as t goes to 0.