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

Abstract

We study the Cauchy problem for the one-dimensional wave equation \[ \partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0. \] The potential $V$ is assumed to be smooth with asymptotic behavior \[ V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty. \] We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field $t\partial_t+x\partial_x$, where the latter are obtained by employing a vector field method on the ``distorted'' Fourier side. 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"

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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) (mɪŋˈkɔːfski,_-ˈkɒf-) combines inertial space and time manifolds (x,y) with a non-inertial reference frame of space and time (x',t')

Wave equation

The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields – as they occur in classical physics –

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transfo