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≥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.Comment: 22paged, 1 figure