Hi, I'm taking a course in partial differential equations.

I need help with this exercise.

Let u(x, t) solves utt - Uxx = 0 with initial conditions u(x, 0) = (x) and ut(x, 0) = 0. Here o(x) is a smooth function that vanishes outside a bounded interval, say [a, b]. Show that lim t- too JR uz dx and lim t- tooJR u, dx both exist and they are equal. Represent their values in terms of o