Blow up for small-amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds

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 $a\mathrm{\mid }{u}_t{\mathrm{\mid }}^p+b\mathrm{\mid }u{\mathrm{\mid }}^q$, 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.