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Yes, it is hyperbolic. If you think of /x=X and /t=T, the equation looks like (X2T2)u=0, and this looks like the equation of a hyperbola.
So for instance, Laplaces equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic. It is useful to classify equations because the solution techniques, and properties of the solutions are different, depending on whether the equation is elliptic, parabolic, or hyperbolic.
The wave equation utt uxx = 0 is hyperbolic. The Laplace equation uxx + uyy = 0 is elliptic. The heat equation ut uxx = 0 is parabolic.
The wave equation is a linear second-order partial differential equation which describes the propagation of oscillations at a fixed speed in some quantity y: A solution to the wave equation in two dimensions propagating over a fixed region [1].
ing to classical non-relativistic Schrdinger equation, any local perturbation of wave function instantaneously affects all infinite region, because this equation is of parabolic type, and its solutions demonstrate infinite speed of perturbations propagation.
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