Does $\square u \pm |u|^pu = 0$ always depend on sign for all $p$?

Question

I remember reading somewhere that the general behavior of nonlinear equations $\square u \pm |u|^pu = 0$ always depend on the sign, that a $+$ sign makes the operator scatter initial data, and the $-$ sign makes it focus initial data. I know it is true in some cases, but is is always true for all values of $p$?

Note: $\square = \frac{\partial^2 }{\partial t^2} - \Delta$