A construction of blow-up solutions for co-rotational wave maps.

摘要

The existence of co-rotational finite time blow up solutions to the wave map problem from R2+1 → N, where N is a surface of revolution with metric drho2 + g(rho)2dtheta 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 - utt + urr + r -1ur = r -2g(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 → 0.