Stability of a class of semilinear waves in $2+1$ dimension under null condition

We will show that in ${\text{RR}}^{2+1}$ semilinear wave equations of the form $-\Box u = u Q(\del u; \del u)$ possess global-in-time solutions if the null condition on $Q(\text{del}u;\text{del}u)$ is assumed. As a consequence, we also provide a new proof, after \cite{Wong}, on the small data global solutions to the wave map equation in ${\text{RR}}^{2+1}$ and no compactness assumptions on the initial data are needed.