The blow-up problem for exponential nonlinearities

Abstract

We give a solution of below-up problem for equation , with data close to constans, in any number of spce 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 with Cauchy data on t = 1 close to (ln2, -2) in s is a large enough integer, must blow-up on a space like hypersurface defined by an equation t = ψ(x) with . Furthermore, the solution has an asymptotic expansion where T = t - ψ(x), valid upto order s - 151 - 10[n/2]. Logarithmic terms are absent if and only if the blow-up surface has vanishing scalar curvature. The blow-up time can be identified with the infimum of the function ψ. Although attention is focused on one equation, the strategy is quite general; it consists in applying the Nash-Moser IFT to a map “singularity data ” to Cauchy data.