Type: Article

Final state problem for the cubic nonlinear Klein-Gordon equation

Nakao Hayashi; Pavel I. Naumkin;

Abstract:

We study the final state problem for the nonlinear Klein-Gordon equation, u(tt)+u - u(xx)= mu u(3), t is an element of R, x is an element of R, where mu is an element of R. We prove the existence of solutions in the neighborhood of the approximate solutions 2 Re U(t)w+ (t), where U(t) is the free evolution group defined by U(t)= F(-1)e(-it)<xi > F, < x > = root 1+ x(2), F and F(-1) are the direct and inverse Fourier transformations, respectively, and w(+) (t, x) = F(-1)((u) over cap (+)(xi)e((3/2)i mu <xi > 2)vertical bar(u) over cap (+)(xi)(2)log (t)), with a given final data u(+) is a real- valued function and parallel to <xi > 3 < i partial derivative >(u) over cap (+)(xi)parallel to L(infinity) is small.