  In this work, we investigate the problem of finite time blow up as well as
the upper bound estimates of lifespan for solutions to small-amplitude
semilinear wave equations with mixed nonlinearities $a |u_t|^p+b |u|^q$, posed
on asymptotically Euclidean manifolds, which is related to both the Strauss
conjecture and Glassey conjecture. In some cases, we obtain existence results,
where the lower bound of the lifespan agrees with the upper bound in order. In
addition, our results apply for semilinear damped wave equations, when the
coefficient of the dissipation term is integrable (without sign condition) and
space-independent.
