Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions

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March 2012, 11(2): 547-556. doi: 10.3934/cpaa.2012.11.547

Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions

Jason Metcalfe ^1, and Jacob Perry ^1,

1.	Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, United States

Received September 2010 Revised February 2011 Published October 2011

Abstract
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This article focuses on proving global existence for quasilinear wave equations with small initial data in homogeneous waveguides with infinite base of dimensions $n\geq 4$. The key estimate is a localized energy estimate for a perturbed wave equation.

Keywords: localized energy estimate., quasilinear wave equation, Waveguides.

Mathematics Subject Classification: Primary: 35L70, 35L2.

Citation: Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure and Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547

References:

[1]	M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math., 87 (2002), 265-279. doi: 10.1007/BF02868477.
[2]	M. Keel, H. F. Smith and C. D. Sogge, Global existence for a quasilinear wave equation outside of star-shaped domains, J. Funct. Anal., 189 (2002), 155-226. doi: 10.1006/jfan.2001.3844.
[3]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. doi: 10.1002/cpa.3160380305.
[4]	S. Klainerman, The null condition and global existence to nonlinear wave equations, Lect. Appl. Math., 23 (1986), 293-326.
[5]	H. Koch, Mixed problems for fully nonlinear hyperbolic equations, Math. Z., 214 (1993), 9-42. doi: 10.1007/BF02572388.
[6]	P. H. Lesky and R. Racke, Nonlinear wave equations in infinite waveguides, Comm. Partial Differential Equations, 28 (2003), 1265-1301. doi: 10.1081/PDE-120024363.
[7]	J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209. doi: 10.1137/050627149.
[8]	J. Metcalfe, C. D. Sogge and A. Stewart, Nonlinear hyperbolic equations in infinite homogeneous waveguides, Comm. Partial Differential Equations, 30 (2005), 643-661. doi: 10.1081/PDE-200059267.
[9]	J. Metcalfe and A. Stewart, Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions, Trans. Amer. Math. Soc., 360 (2008), 171-188. doi: 10.1090/S0002-9947-07-04290-0.
[10]	C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A., 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.
[11]	J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, with an appendix by I. Rodnianski, Int. Math. Res. Not., 2005, 187-231. doi: 10.1155/IMRN.2005.187.
[12]	A. Stewart, "Existence Theorems for Some Nonlinear Hyperbolic Equations on a Waveguide," Ph. D. thesis, Johns Hopkins University, 2006.

show all references

References:

[1]	M. Keel, H. F. Smith and C. D. Sogge, Almost global existence for some semilinear wave equations, J. Anal. Math., 87 (2002), 265-279. doi: 10.1007/BF02868477.
[2]	M. Keel, H. F. Smith and C. D. Sogge, Global existence for a quasilinear wave equation outside of star-shaped domains, J. Funct. Anal., 189 (2002), 155-226. doi: 10.1006/jfan.2001.3844.
[3]	S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38 (1985), 321-332. doi: 10.1002/cpa.3160380305.
[4]	S. Klainerman, The null condition and global existence to nonlinear wave equations, Lect. Appl. Math., 23 (1986), 293-326.
[5]	H. Koch, Mixed problems for fully nonlinear hyperbolic equations, Math. Z., 214 (1993), 9-42. doi: 10.1007/BF02572388.
[6]	P. H. Lesky and R. Racke, Nonlinear wave equations in infinite waveguides, Comm. Partial Differential Equations, 28 (2003), 1265-1301. doi: 10.1081/PDE-120024363.
[7]	J. Metcalfe and C. D. Sogge, Long time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal., 38 (2006), 188-209. doi: 10.1137/050627149.
[8]	J. Metcalfe, C. D. Sogge and A. Stewart, Nonlinear hyperbolic equations in infinite homogeneous waveguides, Comm. Partial Differential Equations, 30 (2005), 643-661. doi: 10.1081/PDE-200059267.
[9]	J. Metcalfe and A. Stewart, Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions, Trans. Amer. Math. Soc., 360 (2008), 171-188. doi: 10.1090/S0002-9947-07-04290-0.
[10]	C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. Ser. A., 306 (1968), 291-296. doi: 10.1098/rspa.1968.0151.
[11]	J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, with an appendix by I. Rodnianski, Int. Math. Res. Not., 2005, 187-231. doi: 10.1155/IMRN.2005.187.
[12]	A. Stewart, "Existence Theorems for Some Nonlinear Hyperbolic Equations on a Waveguide," Ph. D. thesis, Johns Hopkins University, 2006.

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