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- A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials
A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials
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Presents the Cauchy problem for the one-dimensional wave equation ∂ 2 t u (t , x) − ∂ 2 x u (t , x) V (x)u (t , x) = 0. The potential V is assumed to be smooth with asymptotic behavior V (x) ∼ − 1 4|x|−2 as|x|→∞. They 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.
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