  We study a semilinear equation with derivatives satisfying a null condition
on slowly rotating Kerr spacetimes. We prove that given sufficiently small
initial data, the solution exists globally in time and decays with a
quantitative rate to the trivial solution. The proof uses the robust vector
field method. It makes use of the decay properties of the linear wave equation
on Kerr spacetime, in particular the improved decay rates in the region
$\{r\leq \frac{t}{4}\}$.
