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Zbl 0840.35059
Alinhac, S.
Exact life span and geometric explosion for quasilinear hyperbolic systems in space dimension one. (Temps de vie précisé et explosion géométrique pour des systèmes hyperboliques quasilinéaires en dimension un d'espace.) (French)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 22, No.3, 493-515 (1995). ISSN 0391-173X

The author considers the Cauchy problem for hyperbolic systems with two or more unknowns. It is a classical result of {\it F. John} [Commun. Pure Appl. Math. 27, 377-405 (1974; Zbl 0302.35064)] that at smooth initial data, the smooth solutions of a genuinely nonlinear hyperbolic system may breakdown after a finite time. In this paper, the author studies in details the mechanism of the breakdown. In a previous author's paper the concept of ``geometric blow-up'' is introduced. There, the following conjecture is made: a typical blow-uping solution in a point in coincides for $t< T$ near by $m$ with one geometric blowed solution. He proves this conjecture in two cases: (i) System of two equations with two unknown functions. It is supposed in this case that the system is diagonalizable with Riemann invariants. Then, it is proved that the gradient of one of the invariants blows-up. (ii) System of three equations at small initial data. This case was treated by F. John.
[L.G.Vulkov (Russe)]

MSC 2000:: *35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions of PDE

Keywords: geometric blow-uping

Citations: Zbl 0302.35064

Cited in: Zbl 1064.35110

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