  In this paper, we investigate the problem of blow up and sharp upper bound
estimates of the lifespan for the solutions to the semilinear wave equations,
posed on asymptotically Euclidean manifolds. Here the metric is assumed to be
exponential perturbation of the spherical symmetric, long range asymptotically
Euclidean metric. One of the main ingredients in our proof is the construction
of (unbounded) positive entire solutions for
$\Delta_{g}\phi_\lambda=\lambda^{2}\phi_\lambda$, with certain estimates which
are uniform for small parameter $\lambda\in (0,\lambda_0)$. In addition, our
argument works equally well for semilinear damped wave equations, when the
coefficient of the dissipation term is integrable (without sign condition) and
space-independent.
