One dimensional behavior of singular N dimensional solutions of semilinear heat equations

We consider u (x, t) a solution of u(t) = Deltau + \u\(p-1) u that blows up at time T, where u : R-N x [0, T) --> R, p > 1, (N - 2)p < N + 2 and either u(0) greater than or equal to 0 or (3N - 4)p < 3N + 8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T - t --> 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N - I dimensional, we escape logarithmic scales of the variable T - t and give a sharper expansion of the solution with the much smaller error term (T - t)(1/2-eta) for any eta > 0. In particular, if in addition p > 3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C1.1/2-eta for any eta > 0.