ANNULLATO It is believed or conjectured that the semilinear wave equations with scattering space dependent damping admit the Strauss critical exponent. In this work, we are devoted to showing the conjecture is true at least when the decay rate of the space dependent variable coefficients before the damping is larger than 2. Also, if the nonlinear term depends only on the derivative of the solution, we may prove the upper bound of the lifespan is the same as that of the solution of the corresponding problem without damping. This shows in another way the "hyperbolicity" of the equation. This is a joint work with Z.H. Tu.