The weak null conjecture for differentiated semilinear wave systems

Let an asymptotic system correspond to a system of wave equations containing only differentiated semilinear terms. Suppose that the asymptotic system admits global solutions for sufficiently small initial data, and that these solutions grow no faster than

$$r\partial_v\phi\sim r^{c\epsilon}.$$

Weak null conjecture. The original system of wave equations also admits global solutions for sufficiently small initial data.

This conjecture proposes a sufficient refinement of the weak null condition: global existence for the asymptotic system is supplemented by a bound on the growth of its solutions. The supplied text gives no resolution, so its general validity remains open.