energy integrals

[Shaghayegh A]

Problem 4 of the 2012 final exam: http://forum.math.toronto.edu/index.php?topic=177.0
It asks to prove that the total energy (kinetic + potential) is constant with time. I get up to $$\frac{d}{dt} k(t) + p(t) = u_x(\infty) u_t(\infty) - u_x(-\infty) u_t(-\infty) $$ How do I prove $$u_x(\infty) u_t(\infty) - u_x(-\infty) u_t(-\infty) = 0 $$ using the boundary conditions? right now I know $u_t =0$ only when t = 0 (and x is large)

[Victor Ivrii]

Take values at infinity equal 0 (we assume that solution decays there)