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

Abstract

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.

Well-posedness of the equation for the three-form field in eleven-dimensional supergravity

“…This proposition is proven in [10, Proposition 3.1]. Although [10] proves it with zero Cauchy data, the estimate with non-zero Cauchy data is proven in the same manner.…”

“…Thus we see that u λ for λ = 0 evolve under a non-linear wave equation, while u λ for λ = 0 evolve under a non-linear Klein-Gordon equation. This analysis follows the ideas of Metcalfe, Sogge and Stewart [9] and Metcalfe and Stewart [10], who analyze the wave equation on R n+1 ×D, where D is a bounded domain in R m with various boundary conditions. Their analysis splits the function to eigenfunctions of the Laplacian on D with appropriate boundary conditions.…”

“…The author would like to thank his doctoral adviser Daniel Tataru for suggesting the problem and several key ideas in the proof, Jason Metcalfe for discussing the technical aspects of [9] and [10], Robin Graham for exposing the author to questions of supergravity and explaining the results of conformal geometry, Jon Aytac for pointing to the work of Choquet-Bruhat [1] and Tobias Schottdorf for helping improve the exposition.…”

“…Our methods are inspired by the work of Metcalfe, Sogge and Stewart [9] and Metcalfe and Stewart [10] who analyze the quasi-linear wave equation on R 3+1 ×D, where D is a bounded domain in R n with various non-linearities and boundary conditions. Their results do not cover the case in study but we employ some of their ideas in this work.…”

Global stability of spacetimes with supersymmetric compactifications

“…A decomposition of this type has previously been used in the analysis of wave guides, where K is replaced by a compact subset of R d with Neumann boundary conditions, see e.g. [MSS05,MS08]. When applying the vectorfield method to the wave and Klein-Gordon equations, there is a unified approach using a basic energy of the form n i=0 |∂ i h| 2 + λ|h| 2 dµ that can be strengthened by commuting the equation with Γ, the set of generators of translations, rotations, and boosts.…”