INFINITE TIME BLOW-UP FOR HALF-HARMONIC MAP FLOW FROM R INTO S1

We study infinite time blow-up phenomenon for the half-harmonic map flow {0, 1) {u(t) = -(-Delta)(1/2)u+(1/2 pi integral(R)vertical bar u(x) - u(s)(2)/vertical bar x - s vertical bar(2)ds)u in R x (0, infinity), u(., 0) = u(0 )in R, for a smooth function u: R x [0, infinity) -> S-1. Let q(1), ..., q(k) he distinct points in R, there exist a smooth initial datum u(0) and smooth functions xi(j)(t) -> q(j), 0 < mu(j) (t) -> 0, as t -> +infinity, j = 1 ,..., k, such that there exists a smooth solution u(q) of Problem (0.1) of the form u(q) = omega(infinity) + Sigma(k)(j=1) (omega(x-xi(j)(t)/mu(j)(t)) - omega(infinity)) + theta(x, t), where omega is the canonical least energy half-harmonic map, omega(infinity) = ((1)(0)), theta(x, t) -> 0 as t -> +infinity, uniformly away from the points q(j). In addition, the parameter functions mu(j)(t) decay to 0 exponentially.