Critical Blowup for Systems of Semilinear Wave Equations in Low Space Dimensions

We consider the Cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data. Del Santo et al. [“Geometric Optics and Related Topics” (F. Colombini and N. Lerner, Eds.), Progress in Nonlinear Differential Equations and Their Applications, Vol. 32, pp. 117-140, Birkhauser, Boston, 19971 have studied the existence and nonexistence of global classical solutions of the Cauchy problem except for the critical case. In this paper we study the critical case, and we show the nonexistence of global classical solutions and also give the upper bounds of the life span.