Decomposition of global solutions for a class of nonlinear wave equations

In the present paper, we consider global solutions of a class of nonlinear wave equations of the form

$$\begin{aligned} \Box u= N(x,t,u)u, \end{aligned}$$

where the nonlinearity N(x, t, u)u is assumed to satisfy appropriate boundedness assumptions. Under these appropriate assumptions, we prove that the free channel wave operator exists. Moreover, if the interaction term N(x, t, u)u is localized, then we prove that the global solution of the full nonlinear equation can be decomposed into a ‘free’ part and a ‘localized’ part.