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Differential equations, Parabolic

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Differential equations, Parabolic.

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The authors 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 -frac14 |x|^{-2}mbox{ as } |x|o infty. They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field tpartial_t+xpartial_x, where the latter are obtained by employing a vector field method on the âeoedistortedâe Fourier side. In addition, they prove local energy decay estimates. Their 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, âeoeCodimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski spaceâe, preprint arXiv:1310.5606 (2013).

Wave equation. --- Differential equations, Parabolic.

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Evolution equations --- Numerical solutions --- Differential equations [Parabolic] --- Evolution equations - Numerical solutions. --- Differential equations, Parabolic - Numerical solutions. --- Differential equations, Parabolic --- Evolution equations - Numerical solutions --- Differential equations, Parabolic - Numerical solutions

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Differential equations, Parabolic. --- Stochastic partial differential equations.