## 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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