The Existence of Global Solutions to Systems of Quasilinear Wave Equations with Quadratic Nonlinearities in 2-Dimensional Space

Abstract

We deal with systems of quasilinear wave equations which contain quadratic nonlinearities in 2-dimensional space. We have already known that such the system has a smooth solution till the time t0 = Ce-2 for sufficiently small e > 0, where e is the size of initial data. In this paper, we shall show that if quadratic and cubic nonlinearities satisfy so-called Null-condition, then the smooth solution exists globally in time. In the proof of the theorem, we use the Alinhac ghost weight energy.