An elementary proof of global or almost global existence for quasi-linearwave equations

Tohoku Mathematical Journal

Volumn 56, Issue 2, 2004, Pages 271-287

  Hidano, Kunio ^a  

^a MIE UNIVERSITY   (Japan)

Author keywords

Initial value problem; Quasi linear wave equation

Indexed keywords

EID: 3142700008     PISSN: 00408735     EISSN: None     Source Type: Journal    
DOI: 10.2748/tmj/1113246554     Document Type: Article

References (29)

1. R. AGEMI, Global existence of nonlinear elastic waves, Invent. Math. 142 (2000), 225-250.
2. R. AGEMI and K. YOKOYAMA, The null condition and global existence of solutions to systems of wave equations with different speeds. Advances in Nonlinear Partial Differential Equations and Stochastics, Ser. Adv. Math. Appl. Sci., vol. 48, World Sci. Publishing, River Edge, NJ, 1998, pp 43-86.
3. Y. M. CHEN and T. T. Li, Global classical solutions for nonlinear evolution equations, Pitman Monogr. Surveys Pure Appl. Math. 45, Longman Scientific and Technical, Harlow, 1992.
4. K. HIDANO, Small data scattering and blow-up for awave equation with a cubic convolution, Funkcial. Ekvac. 43 (2000), 559-588.
5. K. HIDANO, Scattering problem for the nonlinear wave equation in the finite energy and conformal charge space, J. Funct. Anal. 187 (2001), 274-307.
6. K. HIDANO, The global existence theorem for quasi-linear wave equations with multiple speeds, to appear in Hokkaido Math. J.
7. L. HORMANDER, Lectures on nonlinear hyperbolic differential equations, Math. Appl. 26, Springer, Paris, 1997.
8. A. HOSHIGA and H. KUBO, Global small amplitude solutions of nonlinear hyperbolic systems with a critical exponentunder the null condition, SIAM J. Math. Anal. 31 (2000), 486-513.
9. F. JOHN, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math. 34 (1981), 29-51.
10. F. JOHN, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 (1988), 615-666.
11. F. JOHN, Nonlinear Wave Equations, Formations of Singulalities, Univ. Lecture Ser. 2, Amer. Math. Soc., Providence, RI, 1990.
12. F. JOHN and S. KLAINERMAN, Almost global existence to nonlinear wave equations in three space dimensions, Comm. Pure Appl. Math. 37 (1984), 443-455.
13. M. KEEL, F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math. 87 (2002), 265-279.
14. M. KEEL, F. Smith and C. D. Sogge, Almost global existence for quasilinear wave equations in three space dimensions, J. Amer. Math. Soc. 17 (2004), 109-153.
15. S. KLAINERMAN, Uniform decay estimate and the Lorentz invariance of the classical wave equations, Comm. Pure Appl. Math. 38 (1985), 321-332.
16. 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), 293-326, Lectures in Appl. Math. 23, Amer. Math. Soc., Providence, RI, 1996.
17. 1, Comm. Pure Appl. Math. 40 (1987), 111-117.
18. S. KLAINERMAN and T. C. SIDERIS, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 (1996), 307-321.
19. M. KOVALYOV, Resonance-type behaviour in a system of nonlinear wave equations, J. Differential Equations 77 (1989), 73-83.
20. J. L. METCALFE, Global existence for semilinear wave equations exterior to nontrapping obstacles, Houston J. Math. 30 (2004), 259-281.
21. K. NAKANISHI, Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity, Comm. Partial Differential Equations 24 (1999), 185-221.
22. R. RACKE, Lectures on nonlinear evolution equations, Initial value problems, Aspects Math. E19, Vieweg &Sohn, Braunschweig, 1992.
23. T. C. SIDERIS, The null condition and global existence of nonlinear elastic waves, Invent. Math. 123 (1996), 323-342.
24. T. C. SIDERIS, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2) 151 (2000), 849-874.
25. T. C. SIDERIS and S.-Y. TU, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal. 33 (2002), 477-488.
26. C. D. SOGGE, Lectures on nonlinear wave equations, Monogr. Anal. II, Int. Press, Boston, MA, 1995.
27. C. D. SOGGE, Global existence for nonlinear wave equations with multiple speeds, Harmonic Analysis at Mount Holyoke (South Hadley, MA, 2001), 353-366, Contemp. Math. 320. Amer. Math. Soc., Providence, RI, 2003.
28. W. A. STRAUSS, Nonlinear wave equations, CBMS Reg. Conf. Ser. Math. 73., Amer. Math. Soc., Providence, RI, 1989.
29. K. YOKOYAMA, Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan 52 (2000), 609-632