A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials

作者：Donninger Roland^*; Krieger Joachim

来源：Memoirs of the American Mathematical Society, 2016, 241(1142): 1-+.

DOI：10.1090/memo/1142

We study the Cauchy problem for the one-dimensional wave equation.
2 partial derivative(2)(t)u(t, x) - partial derivative(2)(x)u(t, x) + V (x) u (t, x) = 0.
The potential V is assumed to be smooth with asymptotic behavior
V (x) similar to -1/4 vertical bar x vertical bar(-2) as vertical bar x vertical bar -> infinity.
We derive dispersive estimates, energy estimates, and estimates involving the scaling vector field t partial derivative(t) + x partial derivative(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).

• 出版日期2016-5