## 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

### Keywords:

generic condition on the Cauchy data

Zbl 1080.35043
Full Text:

### References:

 [1] Alinhac, S., Existence d’ondes de raréfaction pour des systèmes quasi-linéaires hyperboliques multidimensionnels.Comm. Partial Differential Equations, 14 (1989), 173–230. · Zbl 0692.35063 [2] –, Approximation près du temps d’explosion des solutions d’équations d’ondes quasilinéaires en dimension deux.Siam J. Math. Anal., 26 (1995), 529–565. · Zbl 0870.35063 [3] –, Temps de vie et comportement explosif des solutions d’équations d’ondes quasi-linéaires en dimension deux, II.Duke Math. J., 73 (1994), 543–560. · Zbl 0844.35102 [4] –, Explosion géométrique pour des systèmes quasi-linéaires.Amer. J. Math., 117 (1995), 987–1017. · Zbl 0840.35060 [5] –, Explosion des solutions d’une équation d’ondes quasi-linéaire en deux dimensions d’espace.Comm. Partial Differential Equations, 21 (1996), 923–969. · Zbl 0858.35082 [6] Alinhac, S., Blowup of small data solutions for a quasilinear wave equation in two space dimensions. To appear inAnn. of Math. · Zbl 1080.35043 [7] –,Blowup for Nonlinear Hyperbolic Equations, Progr. Nonlinear Differential Equations Appl., 17. Birkhäuser Boston, Boston, MA, 1995. [8] Alinhac, S., Stability of geometric blowup. Preprint, Université Paris-Sud, 1997. · Zbl 0896.35087 [9] Alinhac, S. &Gérard, P.,Opérateurs pseudo-différentiels et théorème de Nash-Moser. InterEditions, Paris, 1991. [10] Hörmander, L., The lifespan of classical solutions of nonlinear hyperbolic equations, inPseudodifferential Operators (Oberwolfach, 1986), pp. 214–280. Lecture Notes in Math., 256. Springer-Verlag, Berlin-New York, 1986. [11] –,Lectures on Nonlinear Hyperbolic Differential Equations, Math. Appl., 26. Springer-Verlag, Berlin, 1997. · Zbl 0881.35001 [12] John, F.,Nonlinear Wave Equations, Formation of Singularities. Univ. Lecture Ser., 2. Amer. Math. Soc., Providence, RI, 1990. [13] Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation.Comm. Pure Appl. Math., 38 (1985), 321–332. · Zbl 0635.35059 [14] –, The null condition and global existence to nonlinear wave equations, inNonlinear Systems of Partial Differential Equations in Applied mathematics, Part 1 (Santa Fe, NM, 1984), pp. 293–326. Lectures in Appl. Math., 23. Amer. Math. Soc., Providence, RI, 1986. [15] Majda, A.,Compressible Fluid Flow and Systems of Gonservation Laws in Several Space Variables. Appl. Math. Sci., 53. Springer-Verlag, New York-Berlin, 1984. · Zbl 0537.76001
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