Abstract. This paper deals with the Cauchy problems of nonlinear hyperbolic systems in two space dimensions with small data. We assume that the propagation speeds differ from each other and that nonlinearities are cubic. Then it will be shown that if the nonlinearities satisfy the null condition, there exists a global smooth solution. To prove this kind of claim, one usually makes use of the generalized differential operators Ω ij , S, and L i , which will be introduced in section 1. But it is difficult to adopt the operators L i = x i ∂t + t∂x i to our problem, because they do not commute with the d'Alembertian whose propagation speed is not equal to one. We succeed in taking L i away from the proof of our theorem. One can apply our method to a scalar equation; hence L i are needless in this kind of argument.

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 t 0 ¼ Ce À2 for su‰ciently 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.

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In this paper, we study the behaviour of solutions to quasilinear wave equations in two space dimensions. We obtain blow-up results near the wave front. More precisely, any radially symmetric solution with small initial data is shown to develop singularities in the second order derivatives in finite time, while the first order derivatives and itself remain small. Moreover, we succeed to represent the solution explicitely near the blowing up point.