This paper is concerned with the final value problem for a system of semilinear wave equations. The main issue is to solve the problem when the nonlinearity is of a long-range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence, the generalized wave operator and long range-scattering operator can be constructed.
"Generalized wave operators for a system of semilinear wave equations in three space dimensions." Hokkaido Math. J. 42 (1) 81 - 111, February 2013. https://doi.org/10.14492/hokmj/1362406640