We study the final state problem for the nonlinear Klein-Gordon equation, utt +u- uxx =μ u3, t ∈ R,x ∈ R, where μR. 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 〈φ〉 F, 〈x〉 = 1+ x2, F and F-1 are the direct and inverse Fourier transformations, respectively, and w+ (t,x) = F-1 (û+ (φ) e(3/2) iμ 〈φ〉2 u+ (φ) 2 log t), with a given final data u+ is a real-valued function and ∥ 〈φ〉3 〈 i∥φ〉 u + (φ) ∥ L∞ is small.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics