Dias, João-Paulo and Figueira, Mário:
Blow-up and global existence of a weak solution for a sine-Gordon type quasilinear wave equation
Bollettino dell'Unione Matematica Italiana Serie 8 3-B (2000), fasc. n.3, p. 739-750, Unione Matematica Italiana (English)
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Sunto
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.
Referenze Bibliografiche
[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 1409543 |
Zbl 0857.35085[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 1386463 |
Zbl 0857.35084[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 1648233 |
Zbl 0921.35111[5]
R. J. DIPERNA,
Convergence of approximate solutions to conservation laws,
Arch. Rat. Mech. Anal.,
82 (
1983), 27-70. |
MR 684413 |
Zbl 0519.35054[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 52666 |
Zbl 0047.09101[7]
P. HARTMAN-
A. WINTER,
On hyperbolic differential equations,
Amer. J. Math.,
74 (
1952), 834-864. |
MR 51413 |
Zbl 0048.33302[8]
P. D. LAX,
Development of singularities of solutions of nonlinear hyperbolic partial differential equations,
J. Math. Phys.,
5 (
1964), 611-613. |
MR 165243 |
Zbl 0135.15101[9]
A. MAJDA,
Compressible fluid flow and systems of conservation laws in several space variables,
Applied Math. Sciences, Vol.
53,
Springer,
1984. |
MR 748308 |
Zbl 0537.76001[11]
J. RAUCH,
Partial differential equations,
Graduate texts in Mathematics, Vol.
128,
Springer,
1991. |
MR 1223093 |
Zbl 0742.35001[12]
J. SMOLLER,
Shock waves and reaction-diffusion equations,
Grund. math. Wissenschaften, Vol.
258,
Springer,
1983. |
MR 688146 |
Zbl 0508.35002[13]
L. TARTAR,
Compensated compactness and applications to partial differential equations,
Heriot-Watt Sympos., IV,
Pitman, New York,
1979, 136-212. |
MR 584398 |
Zbl 0437.35004