The blow-up problem for exponential nonlinearities

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: there exists a blow-up surface, near which the solution has logarithmic behavior; its smoothness is estimated in terms of the smoothness of the 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 hypersurface 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 only if the blowup surface has vanishing scalar curvature. The blow-up time can be identified with the infimum of the function psi. Although attention is focused on one equation, the strategy is quite general; it consists in applying the Nash-Moser IFT to a map from ''singularity data'' to Cauchy data.