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Blow-up of classical solutions of quasilinear wave equations in two space dimensions. (Explosion de solutions classiques d’équations d’ondes quasi-linéaires en deux dimensions d’espace.) (French) Zbl 1078.35523

The author considers the Cauchy problem \[ (\partial^2_t- \Delta_x)u+ \sum g_{ij}(\partial u)\partial^2_{ij} u= 0 \] if \(t> 0\), \(x\in\mathbb{R}^2\), with \(C^\infty_0\) initial data of the form \[ \partial^j_t u=\varepsilon u^j_i(x)+ \varepsilon^2 u^j_2(x)+\cdots,\;j= 0,1, \] (\(\varepsilon\) small), if \(t> 0\). Here \(g_{ij}\) are smooth, \(g_{ij}(0)= 0\). It is assumed that a certain function associated with the initial data has a unique minimum which is negative and nondegenerate.
The author announces precise results on the lifespan, the blow-up of the second derivatives, and the structure of the solution near blow-up. Some ideas of the proofs are discussed.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L15 Initial value problems for second-order hyperbolic equations
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