# American Institute of Mathematical Sciences

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

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

Received  September 2010 Revised  February 2011 Published  October 2011

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.
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.

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##### 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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