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Article Dans Une Revue Communications in Partial Differential Equations Année : 1996

The blow-up problem for exponential nonlinearities

Satyanad Kichenassamy

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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Dates et versions

hal-00002647 , version 1 (12-03-2018)

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Satyanad Kichenassamy. The blow-up problem for exponential nonlinearities. Communications in Partial Differential Equations, 1996, 21 (1-2), pp.125-162. ⟨10.1080/03605309608821177⟩. ⟨hal-00002647⟩

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