Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions

Authors:
Jason Metcalfe and Ann Stewart

Journal:
Trans. Amer. Math. Soc. **360** (2008), 171-188

MSC (2000):
Primary 35L70, 42B99

Published electronically:
August 16, 2007

MathSciNet review:
2341999

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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.

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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:
https://doi.org/10.1090/S0002-9947-07-04290-0

Received by editor(s):
September 14, 2005

Published electronically:
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

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.