The blow-up problem for exponential nonlinearities
Résumé
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 $H^s(\mathbb R^n)\times H^{s-1}(\mathbb R^n)$,
$s$ is a large enough integer, must blow-up on a space like
hypersurface defined by an equation $t=\psi(x)$ with $\psi\in
H^{s-146-9[n/2]}(\mathbb R^n)$. Furthermore, the solution has an
asymptotic expansion $\ln(2/T^2) + \sum_{j,k} u_{jk}(x) T^{j+k}
(\ln T)^k$, where $T=t-\psi(x)$, valid upto order $s-151-10[n/2]$.
Logarithmic terms are absent if and only if the blow-up 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.
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