The Blow-up Problem for Exponential Nonlinearities
نویسنده
چکیده
We give a solution of the blow-up problem for equation u = e, 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 u = e with Cauchy data on t = 1 close to (ln 2; 2) in H(R) H (R), s is a large enough integer, must blow-up on a space like hypersurface de ned by an equation t = (x) with 2 H 146 (R). Furthermore, the solution has an asymptotic expansion ln(2=T ) + P j;k ujk(x)T (lnT ), where T = t (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 identi ed with the in mum of the function . 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.
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