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where Tg(u) = b..9u + ga,/3 A(u)(u,a, u,.a), and A(u) is the 2nd fundamental form of N in RK. Let u: M x (0, oo)-+ N be a global weak solution to (1.1). (See [St].) Then u is smooth away from a finite number of singular points {(pi, Ti)} C M x (0, oo). Let (p, T) be a singular point. Then we may assume that for some to E (0, T) and a neighborhood U of p, u is smooth on U x [to, T) and u(t) = u(·, t) -+ u(T) = u(·, T) smoothly on U \ {p} as t tends toT from left. Moreover, by choosing a suitable sequence ti -+ T and by rescaling u(ti) properly near p, one can show that there exist finitely many "bubbles" ( i.e. harmonic maps cpj : 8 2 -+ N, 1 ~ j ~ m, ) associated with the sequence {u(ti)}. These bubbles are responsible for the loss of energy and we have

Energy identity for a class of approximate harmonic maps from surfaces