Geodesic
Blow-up and global existence of a weak solution for a sine-Gordon type quasilinear wave equation
Bollettino della Unione matematica italiana, Série 8, 3B (2000) no. 3, pp. 739-750.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Si considera il problema di Cauchy per l'equazione (cf. [1]): $$\phi_{tt}-\phi_{xx}-\phi^{2}_{x}\phi_{xx}+\sin\phi=0 \qquad (x, t)\in\mathbb{R}\times \mathbb{R}_{+}.$$ Nella prima parte di questo articolo si dimostra, per dati iniziali particolari, un risultato di «blow-up» della soluzione classica locale (in tempo), seguendo le idee introdotte in [8], [2] ed [4]. Nella seconda parte, viene utilizzato il metodo di compattezza per compensazione (cf. [13], [10] ed [5]) ed una estensione del principio delle regioni invarianti (cf. [12]) per dimostrare l'esistenza di una soluzione debole globale entropica. 
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Dias, João-Paulo; Figueira, Mário. Blow-up and global existence of a weak solution for a sine-Gordon type quasilinear wave equation. Bollettino della Unione matematica italiana, Série 8, 3B (2000) no. 3, pp. 739-750. http://geodesic.mathdoc.fr/item/BUMI_2000_8_3B_3_a11/

[1] O. M. Braun-Z. Fei-Y. S. Kivshar-L. Vásquez, Kinks in the Klein-Gordon model with anharmonic interatomic interactions: a variational approach, Phys. Letters A, 157 (1991), 241-245.

[2] J. P. Dias-M. Figueira, On the blow-up of the solutions of a quasilinear wave equation with a semilinear source term, Math. Meth. Appl. Sci., 19 (1996), 1135-1140. | MR | Zbl

[3] J. P. Dias-M. Figueira, Existence d'une solution faible pour une équation d'ondes quasi-linéaires avec un terme de source semi-linéaire, C.R. Acad. Sci. Paris, 322, Série I (1996), 619-624. | MR | Zbl

[4] J. P. Dias-M. Figueira-L. Sanchez, Formation of singularities for the solutions of some quasilinear wave equations, Equa. dérivées part. et applic., Articles dédiés à J. L. Lions, Gauthier-Villars, Paris, 1998, 453-460. | MR | Zbl

[5] R. J. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal., 82 (1983), 27-70. | MR | Zbl

[6] A. Douglis, Some existence theorems for hyperbolic systems of partial differential equations in two independent variables, Comm. Pure Appl. Math., 5 (1952), 119-154. | MR | Zbl

[7] P. Hartman-A. Winter, On hyperbolic differential equations, Amer. J. Math., 74 (1952), 834-864. | MR | Zbl

[8] P. D. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Phys., 5 (1964), 611-613. | MR | Zbl

[9] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Math. Sciences, Vol. 53, Springer, 1984. | MR | Zbl

[10] F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 489-507. | fulltext mini-dml | MR | Zbl

[11] J. Rauch, Partial differential equations, Graduate texts in Mathematics, Vol. 128, Springer, 1991. | MR | Zbl

[12] J. Smoller, Shock waves and reaction-diffusion equations, Grund. math. Wissenschaften, Vol. 258, Springer, 1983. | MR | Zbl

[13] L. Tartar, Compensated compactness and applications to partial differential equations, Heriot-Watt Sympos., IV, Pitman, New York, 1979, 136-212. | MR | Zbl