Type: Article

Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions

Nakao Hayashi; Pavel I. Naumkin;

Abstract:

We study asymptotics around the final states of solutions to the nonlinear Klein-Gordon equations with quadratic nonlinearities in two space dimensions (partial derivative(2)(t) - Delta + m(2))u = lambda u(2), (t, x) is an element of R x R(2), where lambda is an element of C. We prove that if the final states u(1)(+) is an element of H(q/q-1)(4-4/q)(R(2)) boolean AND H(5/2.1) (R(2)) boolean AND H(1)(2)(R(2)), u(2)(+) is an element of H(q/q-1)(3-4/q)(R(2)) boolean AND H(3/2.1) (R(2)) boolean AND H(1)(1)(R(2)), and parallel to u(1)(+)parallel to(H12) + parallel to u(2)(+)parallel to(H11) is sufficiently small, where 4 < q < infinity, then there exists a unique global solution u is an element of C ([T, infinity); L(2) (R(2))) to the nonlinear Klein-Gordon equations such that u (t) tends as t -> infinity in the L(2) sense to the solution u(0) (t) = u(1)(+) cos (< i Delta >(m)t) + (< i del >(-1)(m)u(2)(+)) sin (< i del >(m)t) of the free Klein-Gordon equation.