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

Shijie Dong

Published 2019-10-22Version 1

Under the classical null condition, semilinear wave equations with quadratic nonlinearities in $\RR^{2+1}$ might not have global-in-time solutions. We will show that in $\RR^{2+1}$ semilinear wave equations of the form $-\Box u = u Q(\del u; \del u)$ possess global-in-time solutions if the null condition on $Q(\del u; \del u)$ 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 $\RR^{2+1}$ and no compactness assumptions on the initial data are needed.