Abstract
We provide a proof of global existence of solutions to quasilinear wave equations satisfying the null condition in certain exterior domains. In particular, our proof does not require estimation of the fundamental solution for the free wave equation. We instead rely upon a class of Keel–Smith–Sogge estimates for the perturbed wave equation. Using this, a notable simplification is made as compared to previous works concerning wave equations in exterior domains: one no longer needs to distinguish the scaling vector field from the other admissible vector fields.
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References
Agemi, R., Yokoyama, K.: The null condition and global existence of solutions to systems of wave equations with different speeds. Adv. Nonlinear Partial Differ. Equ. Stochastics, 43–86 (1998)
Alinhac, S.: On the Morawetz/KSS inequality for the wave equation on a curved background. preprint (2005)
Christodoulou D. (1986). Global solutions of nonlinear hyperbolic equations for small initial data. Comm. Pure Appl. Math. 39: 267–282
Hidano K. (2004). An elementary proof of global or almost global existence for quasi-linear wave equations. Tohoku Math. J. 56: 271–287
Hidano K. and Yokoyama K. (2005). A remark on the almost global existence theorems of Keel, Smith and Sogge. Funkcial. Ekvac. 48: 1–34
Hörmander L. (1997). Lectures on Nonlinear Hyperbolic Equations. Springer, Heidelberg New York, Berlin
Katayama S. (2004). Global existence for a class of systems of nonlinear wave equations in three space dimensions. Chinese Ann. Math. Ser. B, 25: 463–482
Katayama S. (2005). Global existence for systems of wave equations with nonresonant nonlinearities and null forms. J. Differ. Equ. 209: 140–171
Katayama, S.: A remark on systems of nonlinear wave equations with different propagation speeds. preprint (2003).
Keel M., Smith H. and Sogge C.D. (2002). Global existence for a quasilinear wave equation outside of star-shaped domains. J. Funct. Anal. 189: 155–226
Keel M., Smith H. and Sogge C.D. (2002). Almost global existence for some semilinear wave equations. J. D’Analyse 87: 265–279
Keel M., Smith H. and Sogge C.D. (2004). Almost global existence for quasilinear wave equations in three space dimensions. J. Am. Math. Soc. 17: 109–153
Klainerman S. (1985). Uniform decay estimates and the Lorentz invariance of the classical wave equation. Commun. Pure Appl. Math. 38: 321–332
Klainerman S. (1986). The null condition and global existence to nonlinear wave equations. Lect. Appl. Math. 23: 293–326
Klainerman S. and Sideris T. (1996). On almost global existence for nonrelativistic wave equations in 3d. Commun. Pure Appl. Math. 49: 307–321
Kubota K. and Yokoyama K. (2001). Global existences of classical solutions to systems of nonlinear wave equations with different speeds of propagation. Jpn. J. Math. 27: 113–202
Lax P.D., Morawetz C.S. and Phillips R.S. (1963). Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle. Commun. Pure Appl. Math. 16: 477–486
Metcalfe J. (2004). Global existence for semilinear wave equations exterior to nontrapping obstacles.. Houston J. Math. 30: 259–281
Metcalfe J., Nakamura M. and Sogge C.D. (2005). Global existence of solutions to multiple speed systems of quasilinear wave equations in exterior domains. Forum. Math. 17: 133–168
Metcalfe J., Nakamura M. and Sogge C.D. (2005). Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition. Jpn. J. Math. 31: 391–472
Metcalfe J. and Sogge C.D. (2005). Hyperbolic trapped rays and global existence of quasilinear wave equations. Invent. Math. 159: 75–117
Metcalfe, J., Sogge, C.D.: Global existence for Dirichlet-wave equations with quadratic nonlinearities in high dimensions. Math. Ann. (to appear)
Metcalfe J. and Sogge C.D. (2006). Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods. SIAM J. Math. Anal. 38: 188–209
Morawetz C.S. (1961). The decay of solutions of the exterior initial-boundary problem for the wave equation. Commun. Pure Appl. Math. 14: 561–568
Sideris T. (2000). Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. Math. 151: 849–874
Sideris T. and Tu S.Y. (2001). Global existence for systems of nonlinear wave equations in 3D with multiple speeds. SIAM J. Math. Anal. 33: 477–488
Sogge C.D. (1995). Lectures on Nonlinear Wave Equations. International Press, Cambridge, MA
Sogge, C.D.: Global existence for nonlinear wave equations with multiple speeds. In Harmonic analysis, Mount Holyoke (South Hadley, 2001), pp. 353–366, Contemp. Math., 320, Amer. Math. Soc., Providence, RI, 2003.
Sterbenz, J.: Angular regularity and Strichartz estimates for the wave equation with an appendix by I. Rodnianski. Int. Math. Res. Notes 2005, 187–231.
Strauss W.A. (1975). Dispersal of waves vanishing on the boundary of an exterior domain. Commun. Pure Appl. Math. 28: 265–278
Yokoyama K. (2000). Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions. J. Math. Soc. Japan 52: 609–632
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Metcalfe, J., Sogge, C.D. Global existence of null-form wave equations in exterior domains. Math. Z. 256, 521–549 (2007). https://doi.org/10.1007/s00209-006-0083-2
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DOI: https://doi.org/10.1007/s00209-006-0083-2