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Explosion of a two-dimensional quasilinear wave equation. (Explosion des solutions d’une équation d’ondes quasi-linéaire en deux dimensions d’espace.) (French) Zbl 0858.35082

This paper proves that singularity formation for \(u_{xt}+u_xu_{xx}+ \varepsilon u_{yy}=0\) occurs in the manner predicted by the unfolding method introduced by the author, modulo some “non-degeneracy” assumptions which ensure control of the singularity for \(\varepsilon=0\). One assumes periodicity in \(y\), and one takes the boundary conditions \(u=u_0\) for \(t=0\), \(u=u_x=0\) for \(x=0\). The equation is solved for \(t<\overline{T}\), and the singularity occurs at a single point. This model equation is a transverse perturbation of Burgers’ equation; it also occurs, as the author explains, as a model for more general situations. The method consists in casting the “unfolded system” for \(t<\overline{T}\) into a form suitable for the use of the Nash-Moser implicit function theorem; the lack of well-posedness of the unfolded system is obviated by seeking the solution only on the “good” side of the singularity. It seems that this paper contains several ideas of wider applicability, and is therefore a very significant addition to the literature.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B20 Perturbations in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
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[1] DOI: 10.1080/03605308908820595 · Zbl 0692.35063
[2] Alinhac, S. 1993. Explosion géométrique pour des systèmes quasi-linéaires. Séminaire d’EDP. 1993, Ecole Polytechnique, Paris. et article à paraître dans Amer. J. Math. (1995)
[3] DOI: 10.1215/S0012-7094-94-07322-5 · Zbl 0844.35102
[4] Alinhac S., Progress in Nonlinear Differential Equations and their Applications (1995)
[5] Alinhac S., Opérateurs pseudo-différentiels et théorème de Nash-Moser (1991)
[6] Alinhac S., à paraître dans Annali Sc. Norm. Pisa (1995)
[7] DOI: 10.1090/S0002-9947-1986-0849476-3
[8] DOI: 10.1007/BF00280224 · Zbl 0593.35055
[9] Hörmander L., Lecture Notes Math. 1256 pp 214– (1986)
[10] DOI: 10.1007/BF00251855 · Zbl 0331.35020
[11] John F., University Lecture Series (1990)
[12] Kichenassamy S. : The blow-up problem for exponential nonlinearities, preprint, (1995).
[13] DOI: 10.1002/cpa.3160100406 · Zbl 0081.08803
[14] Majda A., Springer Appl. Math. Sc. 53 (1984)
[15] Smoller J., Grundlehr. 258 (1983)
[16] Strauss W., Conf. Board Math. Sc. 73 (1989)
[17] Kichenassamy S. : The blow-up problem for exponential nonlinearities, preprint, (1995).Zuily C. : Solutions en grand temps d’équations d’ondes non linéaires, Séminaire Bourbaki 779, Paris (1993/1994).
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