We give a solution of the blow-up problem for equation square u = e(u)
, with data. close to constants, in any number of space dimensions: th
ere exists a blow-up surface, near which the solution has logarithmic
behavior; its smoothness is estimated in terms of the smoothness of th
e data. More precisely, we prove that for any solution of square u = e
(u) with Cauchy data on t = 1 close to (ln 2, -2) in H-3(R(n)) x H-s-1
(R(n)), s is a large enough integer, must blow-up on a space like hype
rsurface defined by an equation t = psi(x) with psi epsilon H-s-146-9[
n/2](R(n)). Furthermore, the solution has an asymptotic expansion ln(2
/T-2) + Sigma(j,k) u(jk)(x)T-j+k(ln T)(k), where T = t - psi(x), valid
upto order s - 151 - 10[n/2]. Logarithmic terms are absent if and onl
y if the blowup surface has vanishing scalar curvature. The blow-up ti
me can be identified with the infimum of the function psi. Although at
tention is focused on one equation, the strategy is quite general; it
consists in applying the Nash-Moser IFT to a map from ''singularity da
ta'' to Cauchy data.