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Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions
Author(s):
Jason
Metcalfe;
Ann
Stewart
Journal:
Trans. Amer. Math. Soc.
360
(2008),
171-188.
MSC (2000):
Primary 35L70, 42B99
Posted:
August 16, 2007
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Abstract:
In this paper, we prove almost global existence of solutions to certain quasilinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides with Neumann boundary conditions. We use a Galerkin method to expand the Laplacian of the compact base in terms of its eigenfunctions. For those terms corresponding to zero modes, we obtain decay using analogs of estimates of Klainerman and Sideris. For the nonzero modes, estimates for Klein-Gordon equations, which provide better decay, are available.
References:
-
- 1.
- D. Gilbarg and N. Trudinger: Elliptic partial differential equations of second order, Springer, Second Ed., Third Printing, 1998.
- 2.
- K. Hidano: An elementary proof of global or almost global existence for quasi-linear wave equations. Tohoku Math. J. 56 (2004), 271-287. MR 2053322 (2005d:35175)
- 3.
- L. Hörmander: Lectures on nonlinear hyperbolic equations, Springer-Verlag, Berlin, 1997. MR 1466700 (98e:35103)
- 4.
- F. John: Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34 (1981), 29-51. MR 600571 (83d:35096)
- 5.
- F. John: Nonlinear wave equations, formation of singularities. Amer. Math. Soc., Providence, 1990. MR 1066694 (91g:35001)
- 6.
- M. Keel, H. Smith, and C. D. Sogge: Global existence for a quasilinear wave equation outside of star-shaped domains. J. Funct. Anal. 189 (2002), 155-226. MR 1887632 (2003f:35209)
- 7.
- M. Keel, H. Smith, and C. D. Sogge: Almost global existence for some semilinear wave equations. J. Anal. Math. 87 (2002), 265-279. MR 1945285 (2003m:35167)
- 8.
- S. Klainerman and T. C. Sideris: On almost global existence for nonrelativistic wave equations in 3d. Comm. Pure Appl. Math. 49, (1996), 307-321. MR 1374174 (96m:35231)
- 9.
- P. H. Lesky and R. Racke: Nonlinear wave equations in infinite waveguides. Comm. Partial Differential Equations 28 (2003), 1265-1301. MR 1998938 (2004f:35122)
- 10.
- J. Metcalfe and C. D. Sogge: Hyperbolic trapped rays and global existence of quasilinear wave equations. Invent. Math. 159 (2005), 75-117.
- 11.
- J. Metcalfe, C. D. Sogge, and A. Stewart: Nonlinear hyperbolic equations in infinite homogeneous waveguides. Comm. Partial Differential Equations 30 (2005), 643-661. MR 2153511 (2006b:35230)
- 12.
- T. C. Sideris: Global behavior of solutions to nonlinear wave equations in three dimensions. Comm. Partial Differential Equations 8 (1983), 1291-1323. MR 711440 (85e:35081)
- 13.
- T. C. Sideris: Nonresonance and global existence of prestressed nonlinear elastic waves. Ann. of Math. 151 (2000), 849-874. MR 1765712 (2001g:35180)
- 14.
- C. D. Sogge: Lectures on nonlinear wave equations, International Press, Cambridge, MA, 1995. MR 1715192 (2000g:35153)
- 15.
- M. Taylor: Partial Differential Equations I, Springer-Verlag, New York, 1996. MR 1395148 (98b:35002b)
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Additional Information:
Jason
Metcalfe
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720-3840
Address at time of publication:
Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250
Email:
metcalfe@math.berkeley.edu
Ann
Stewart
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015
Address at time of publication:
Department of Mathematics, Hood College, Frederick, Maryland 21701
DOI:
10.1090/S0002-9947-07-04290-0
PII:
S 0002-9947(07)04290-0
Received by editor(s):
September 14, 2005
Posted:
August 16, 2007
Additional Notes:
The authors were supported in part by the NSF
A portion of this work was completed while the authors were visiting the Mathematical Sciences Research Institute (MSRI). The authors gratefully acknowledge the hospitality and support of MSRI
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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