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A vector field method on the distorted Fourier side and decay for wave equations with potentials


About this Title

Roland Donninger and Joachim Krieger

Publication: Memoirs of the American Mathematical Society
Publication Year: 2016; Volume 241, Number 1142
ISBNs: 978-1-4704-1873-1 (print); 978-1-4704-2877-8 (online)
DOI: http://dx.doi.org/10.1090/memo/1142
Published electronically: December 10, 2015

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Weyl-Titchmarsh Theory for $ A$
  • Chapter 3. Dispersive Bounds
  • Chapter 4. Energy Bounds
  • Chapter 5. Vector Field Bounds
  • Chapter 6. Higher Order Vector Field Bounds
  • Chapter 7. Local Energy Decay
  • Chapter 8. Bounds for Data in Divergence Form

Abstract


We study the Cauchy problem for the one-dimensional wave equation \[ ∂_{t}^{2} u(t,x)-∂_{x}^{2} u(t,x)+V(x)u(t,x)=0. \] The potential $V$ is assumed to be smooth with asymptotic behavior \[ V(x)∼-\tfrac14 |x|^{-}2as |x|→∞. \] We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field $t∂_{t}+x∂_{x}$, where the latter are obtained by employing a vector field method on the “distorted” Fourier side. In addition, we prove local energy decay estimates. Our results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space, see Donninger, Krieger, Szeftel, and Wong, “Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space”, preprint arXiv:1310.5606 (2013).

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