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

The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes