Global oscillatory waves for second order quasilinear wave equations
HTML articles powered by AMS MathViewer
- by Paul Godin PDF
- Trans. Amer. Math. Soc. 346 (1994), 523-547 Request permission
Abstract:
In this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of Cauchy problems for a class of scalar second order quasilinear wave equations, when the space dimension is odd and at least equal to $3$. If time is bounded, corresponding results for quasilinear first order systems were obtained by Guès; to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion. To obtain global results, suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time.References
- Demetrios Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math. 39 (1986), no. 2, 267–282. MR 820070, DOI 10.1002/cpa.3160390205 R. Courant and D. Hilbert, Methods of mathematical physics, Vol. 2, Interscience, 1962.
- Paul Godin, Long time existence of piecewise smooth progressing waves for semilinear wave equations, J. Math. Pures Appl. (9) 72 (1993), no. 1, 15–56. MR 1201252
- Paul Godin, Global sound waves for quasilinear second order wave equations, Math. Ann. 298 (1994), no. 3, 497–531. MR 1262773, DOI 10.1007/BF01459748
- Olivier Guès, Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires, Asymptotic Anal. 6 (1993), no. 3, 241–269 (French, with English summary). MR 1201195 J. L. Joly, G. Métivier, and J. Rauch, Remarques sur l’optique géométrique non linéaire multidimensionnelle, exposé no. 1, Séminaire EDP, Ecole Polytechnique, Palaiseau, 1990-91.
- Sergiu Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43–101. MR 544044, DOI 10.1002/cpa.3160330104
- Sergiu Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math. 38 (1985), no. 3, 321–332. MR 784477, DOI 10.1002/cpa.3160380305
- S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984) Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293–326. MR 837683
- Li Ta-tsien and Chen Yun-mei, Initial value problems for nonlinear wave equations, Comm. Partial Differential Equations 13 (1988), no. 4, 383–422. MR 920909, DOI 10.1080/03605308808820547
- A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308, DOI 10.1007/978-1-4612-1116-7
- Guy Métivier, Problèmes de Cauchy et ondes non linéaires, Journées “Équations aux dérivées partielles” (Saint Jean de Monts, 1986) École Polytech., Palaiseau, 1986, pp. No. I, 29 (French). MR 874543 K. Morawetz, Energy decay for star-shaped obstacles, in Scattering Theory (P. D. Lax and R. Phillips), Academic Press, 1967, pp. 261-264. M. Spivak, A comprehensive introduction to differential geometry, Vols. 3, 4, Publish or Perish, 1979.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 346 (1994), 523-547
- MSC: Primary 35L70; Secondary 35B40, 58G16
- DOI: https://doi.org/10.1090/S0002-9947-1994-1270662-X
- MathSciNet review: 1270662