The null condition for systems of nonlinear wave equations and its related topics

Abstract

In this article, we consider the Cauchy problem for systems of nonlinear wave equations, whose nonlinear terms depend mainly on derivatives of the unknown, with small, smooth, and compactly supported initial data. If the nonlinear terms have the critical power, we need some structural conditions to obtain the small data global existence. The null condition is one such condition, but recently some weaker conditions were also found. We discuss the small data global existence and the asymptotic behavior of solutions under these conditions. The corresponding result for nonlinear Schrödinger equations will be also discussed.