Global existence and blow-up for a hyperbolic system in three space dimensions

Abstract

Using the technique developed by F. Jolm in $\left[7\right]$, we study the existence and the nonexistence of global classical solutions to the Cauchy problem for \[ \partial_{t}^{2}u-\Delta_{x}u=\mid v\mid^{p}, \] \[ \partial_{t}^{2}v-\Delta_{x}v=\mid u\mid^{q}, \] in $\mathbf{R_{\textrm{x}}^{\textrm{3}}}\times[0,+\infty[$