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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

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 & 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.  doi: 10.1007/BF02868477.  Google Scholar

[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.  doi: 10.1006/jfan.2001.3844.  Google Scholar

[3]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321.  doi: 10.1002/cpa.3160380305.  Google Scholar

[4]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, Lect. Appl. Math., 23 (1986), 293.   Google Scholar

[5]

H. Koch, Mixed problems for fully nonlinear hyperbolic equations,, Math. Z., 214 (1993), 9.  doi: 10.1007/BF02572388.  Google Scholar

[6]

P. H. Lesky and R. Racke, Nonlinear wave equations in infinite waveguides,, Comm. Partial Differential Equations, 28 (2003), 1265.  doi: 10.1081/PDE-120024363.  Google Scholar

[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.  doi: 10.1137/050627149.  Google Scholar

[8]

J. Metcalfe, C. D. Sogge and A. Stewart, Nonlinear hyperbolic equations in infinite homogeneous waveguides,, Comm. Partial Differential Equations, 30 (2005), 643.  doi: 10.1081/PDE-200059267.  Google Scholar

[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.  doi: 10.1090/S0002-9947-07-04290-0.  Google Scholar

[10]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations,, Proc. Roy. Soc. Ser. A., 306 (1968), 291.  doi: 10.1098/rspa.1968.0151.  Google Scholar

[11]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, with an appendix by I. Rodnianski,, Int. Math. Res. Not., 2005 (2005), 187.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[12]

A. Stewart, "Existence Theorems for Some Nonlinear Hyperbolic Equations on a Waveguide,", Ph. D. thesis, (2006).   Google Scholar

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.  doi: 10.1007/BF02868477.  Google Scholar

[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.  doi: 10.1006/jfan.2001.3844.  Google Scholar

[3]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation,, Comm. Pure Appl. Math., 38 (1985), 321.  doi: 10.1002/cpa.3160380305.  Google Scholar

[4]

S. Klainerman, The null condition and global existence to nonlinear wave equations,, Lect. Appl. Math., 23 (1986), 293.   Google Scholar

[5]

H. Koch, Mixed problems for fully nonlinear hyperbolic equations,, Math. Z., 214 (1993), 9.  doi: 10.1007/BF02572388.  Google Scholar

[6]

P. H. Lesky and R. Racke, Nonlinear wave equations in infinite waveguides,, Comm. Partial Differential Equations, 28 (2003), 1265.  doi: 10.1081/PDE-120024363.  Google Scholar

[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.  doi: 10.1137/050627149.  Google Scholar

[8]

J. Metcalfe, C. D. Sogge and A. Stewart, Nonlinear hyperbolic equations in infinite homogeneous waveguides,, Comm. Partial Differential Equations, 30 (2005), 643.  doi: 10.1081/PDE-200059267.  Google Scholar

[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.  doi: 10.1090/S0002-9947-07-04290-0.  Google Scholar

[10]

C. S. Morawetz, Time decay for the nonlinear Klein-Gordon equations,, Proc. Roy. Soc. Ser. A., 306 (1968), 291.  doi: 10.1098/rspa.1968.0151.  Google Scholar

[11]

J. Sterbenz, Angular regularity and Strichartz estimates for the wave equation, with an appendix by I. Rodnianski,, Int. Math. Res. Not., 2005 (2005), 187.  doi: 10.1155/IMRN.2005.187.  Google Scholar

[12]

A. Stewart, "Existence Theorems for Some Nonlinear Hyperbolic Equations on a Waveguide,", Ph. D. thesis, (2006).   Google Scholar

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