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Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions. II. (English) Zbl 0973.35135

From the introduction: This work is a continuation of our previous work [Ann. Math. (2) 149, No. 1, 97–127 (1999; Zbl 1080.35043)]. We consider in both quasilinear wave equations in \(\mathbb R^{2+1}\), \[ \partial^2_t u-\Delta_x u+ \sum_{0\leq i,j,k\leq 2} g^k_{ij}\partial_k u\partial^2_{ij} u=0,\tag{1} \] where \(x_0= t\), \(x= (x_1,x_2)\), \(g^k_{ij}= g^k_{ji}\). We assume that the Cauchy data are \(C^\infty\) and small, \(u(x,0)= \varepsilon u^0_1+ \varepsilon^2 u^0_2+\cdots\), \(\partial_t u(x,0)= \varepsilon u^1_1+ \varepsilon^2 u^1_2+\cdots\), and supported in a fixed ball of radius \(M\).
In our previous work [loc. cit.], we were able to prove actual blowup only for the special example of (1), \((\partial^2_t- \Delta)u= (\partial_t u)(\partial^2_t u)\).
In the present work, we prove that actual blowup takes place at the suggested time for a general equation (1). The only assumption we need is a “generic” condition on the Cauchy data.

MSC:

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

Citations:

Zbl 1080.35043
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References:

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