Finite time blow up for systems of wave equations with multiple speeds posed on asymptotically Euclidean manifold

In this work, we study the finite time blow-up phenomenon of three types of semilinear wave systems with multiple speeds, posed on asymptotically Euclidean manifolds. We establish the upper bound estimates for the lifespan of solutions when the spatial dimension $n\ge 2$. In particular, for system related to the Glassey conjecture, we obtained the finite time blow up results under a new curve. This new curve is sharp at least for $n=3$.